diff --git a/Appendices/ComplexNumbers.md b/Appendices/ComplexNumbers.md
index b250e6b548bb9c70cb71e11fe22fe279dc192bdd..757455f008f0ce9a2134f48cd87267d482bfda17 100644
--- a/Appendices/ComplexNumbers.md
+++ b/Appendices/ComplexNumbers.md
@@ -1061,6 +1061,15 @@ The three solutions from {prf:ref}`Ex:ComplexNumbers:threesolutions`.
 
 We can generalize the method for solving $z^n=w$ from the example above:
 
+```{applet}
+:url: appendix/complex_numbers
+:fig: Images/Fig-ComplexNumbers-threesolfig.svg
+:name: Fig:ComplexNumbers:general
+:position: 0, -1
+
+The n solutions from {prf:ref}`Ex:ComplexNumbers:threesolutions` generalized.
+```
+
 ```{prf:algorithm} Solving $z^n=w$
 
 <ol>
diff --git a/Chapter1/Inner_Product.md b/Chapter1/Inner_Product.md
index a296bc6f65429e45f99be5d214b727e756c2c005..29cae4648df6ea0e87b9300457e1d7cbbe0a7d59 100644
--- a/Chapter1/Inner_Product.md
+++ b/Chapter1/Inner_Product.md
@@ -336,9 +336,6 @@ $$
 
 :::
 
-
-
-
 :::{exercise}
 :label: Exc:InnerProduct:(v-w)(v+w)
 
@@ -436,7 +433,7 @@ Two vectors $\mathbf{v}$ and $\mathbf{w}$ in $\mathbb{R}^n$ are called _orthogon
 :::{prf:example}
 :label: Ex:InnerProduct:CheckVectorsOrthogonal
 
-Let $\mathbf{u} = \begin{bmatrix} 1\\2\\-1\\-1\end{bmatrix}$, $\mathbf{v} = \begin{bmatrix} 3\\-1\\2\\-1\end{bmatrix}$ and 
+Let $\mathbf{u} = \begin{bmatrix} 1\\2\\-1\\-1\end{bmatrix}$, $\mathbf{v} = \begin{bmatrix} 3\\-1\\2\\-1\end{bmatrix}$ and
 $\mathbf{w} = \begin{bmatrix} 2\\2\\-1\\2\end{bmatrix}$.
 
 We compute
@@ -458,6 +455,8 @@ $$
 
 and conclude that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, $\mathbf{u}$ and $\mathbf{w}$ are not orthogonal, <br/> $\mathbf{v}$ and $\mathbf{w}$ are orthogonal.
 
+:::
+
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/59912254-6fc8-43c7-9c44-1ea7eab1c236?id=62409
 :label: grasple_exercise_1_2_1T
@@ -531,7 +530,9 @@ $$
 
 :::
 
-:::{figure} Images/Fig-InnerProduct-ProjectionVectorLine.svg
+:::{applet}
+:url: dot_product/innerproduct_projectionvectorline
+:fig: Images/Fig-InnerProduct-ProjectionVectorLine.svg
 :name: Fig:InnerProduct:ProjectionVectorLine
 
 Projection of a vector $\mathbf{w}$ onto a nonzero vector $\mathbf{v}$
@@ -655,7 +656,7 @@ $$
 
 :::
 
-:::{dropdown} Solution to&nbsp;{numref}`Exc:InnerProduct:SameProjectionThenWhat`&nbsp;(_click to show_)
+::::{dropdown} Solution to&nbsp;{numref}`Exc:InnerProduct:SameProjectionThenWhat`&nbsp;(_click to show_)
 
 Suppose $\text{proj}_{\mathbf{v}}(\mathbf{w}_1) = \text{proj}_{\mathbf{v}}(\mathbf{w}_2) $.  Thus  $\dfrac{\mathbf{w}_1\ip\mathbf{v}}{\mathbf{v}\ip\mathbf{v}} \mathbf{v} = \dfrac{\mathbf{w}_2\ip\mathbf{v}}{\mathbf{v}\ip\mathbf{v}} \mathbf{v}$.
 
@@ -696,6 +697,8 @@ Two vectors $\vect{w}_1$, $\vect{w}_2 $  with the same projection onto $\vect{v}
 
 :::
 
+::::
+
 (Subsec:InnerProduct:Norm_in_Rn)=
 
 ## Norm in $\mathbb{R}^n$
@@ -761,8 +764,11 @@ iii. Triangle Inequality:
 
 The first two of these properties are very easy to prove. The proof of the triangle inequality we postpone until the end of the section. {numref}`Figure %s <Fig:InnerProduct:TriangleInequality>` explains the name.
 
-:::{figure} Images/Fig-InnerProduct-TriangleInequality.svg
+:::{applet}
+:url: dot_product/triangle_inequality
+:fig: Images/Fig-InnerProduct-TriangleInequality.svg
 :name: Fig:InnerProduct:TriangleInequality
+:position: 2,2
 
 The Triangle Inequality
 :::
@@ -833,7 +839,6 @@ $$
 The distance between two vectors
 :::
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/5bc4274c-56a0-461b-bd3d-9f8bdb8f44e0?id=69740
 :label: grasple_exercise_1_2_2
@@ -847,7 +852,7 @@ From the rules of the norm the following rules of the distance function can be d
 ::::{prf:proposition}
 :label: Prop:InnerProduct:PropertiesDistance
 
-For any  three vectors $\mathbf{u}, \mathbf{v}$ and $\mathbf{w} \in \mathbb{R}^{n}$ the following statements hold.
+For any three vectors $\mathbf{u}, \mathbf{v}$ and $\mathbf{w} \in \mathbb{R}^{n}$ the following statements hold.
 
 i. $\text{dist}(\vect{u},\vect{v}) = \text{dist}(\vect{v},\vect{u})$;
 
@@ -1008,7 +1013,6 @@ $$
   \iff \mathbf{v}\perp\mathbf{w}.
 $$
 
-
 :::
 
 :::{prf:example}
@@ -1366,7 +1370,7 @@ $$
     \norm{\vect{v}_i - \vect{v}_j} = \sqrt{2^2 + 2^2 + 0^2} = \sqrt{8}, \text{ for } i \neq j.
 $$
 
-The angle between, for instance,  $\mathbf{v}_1$ and $\mathbf{v}_3$ is determined by
+The angle between, for instance, $\mathbf{v}_1$ and $\mathbf{v}_3$ is determined by
 
 $$
    \cos(\varphi) = \dfrac{\mathbf{v}_1\ip\mathbf{v}_3}{\norm{\mathbf{v}_1}\norm{\mathbf{v}_3}} = \dfrac{-1}{\sqrt{3}\cdot\sqrt{3}} = -\frac13.
@@ -1518,7 +1522,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
 :url: https://embed.grasple.com/exercises/67334454-d109-45a2-b640-545041ff896d?id=62416
 :label: grasple_exercise_1_2_13
 :dropdown:
-:description: Find $\text{proj}_{\mathbf{v}}(\mathbf{w})$ in $\R^2$.
+:description: Find $\text{proj}*{\mathbf{v}}(\mathbf{w})$ in $\R^2$.
 
 ::::
 
@@ -1538,8 +1542,6 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
 
 ::::
 
-
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/161ecdf6-4cfb-41ba-bc16-685fe8532471?id=62414
 :label: grasple_exercise_1_2_16
@@ -1560,7 +1562,7 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
 :url: https://embed.grasple.com/exercises/407cb45d-2baf-4b0d-a1eb-6e51186e19f3?id=69738
 :label: grasple_exercise_1_2_18
 :dropdown:
-:description: What to conclude from  &nbsp;$\norm{\vect{v}+\vect{w}} = \norm{\vect{v}}+\norm{\vect{w}}$?
+:description: What to conclude from &nbsp;$\norm{\vect{v}+\vect{w}} = \norm{\vect{v}}+\norm{\vect{w}}$?
 
 ::::
 
@@ -1574,9 +1576,9 @@ we may conclude that for large $n$ in $\mathbb{R}^n$ the two vectors are 'almost
 
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/2a2423c3-0907-40b7-bd5f-7607baf7cc09?id=62668
-:label: grasple_exercise_1_2_20
+:label: grasple*exercise_1_2_20
 :dropdown:
-:description: What to conclude 
-from $\text{proj}_{\mathbf{v}}(\mathbf{w}_1 ) = \text{proj}_{\mathbf{v}}(\mathbf{w}_2)$?
+:description: What to conclude
+from $\text{proj}_{\mathbf{v}}(\mathbf{w}_1 ) = \text{proj}_{\mathbf{v}}(\mathbf{w}\_2)$?
 
 ::::
diff --git a/Chapter2/LinearSystems.md b/Chapter2/LinearSystems.md
index cc742b828fd0a861ca66157a74f50847d60cfcdb..703596e51eb1f2885bdcbf4c6a2df9de96e94e77 100644
--- a/Chapter2/LinearSystems.md
+++ b/Chapter2/LinearSystems.md
@@ -97,7 +97,7 @@ $$
  x_1 = c_1, \,\, x_2 = c_2, \, \ldots\, , \,\, x_n = c_n
 $$
 
-into each of the equations yields a true identity.
+into each of the equations yields true identities.
 
 The **solution set** or **general solution** of a system is the set of all solutions.
 
@@ -303,7 +303,7 @@ $$
 
    \left\{\begin{array}{ccccc}
     2x_1&-&5x_2&=&-2\\
-    4x_1&-&7x_2&=& 2
+    4x_1&-&7x_2&=& 2.
     \end{array}\right.
 $$
 
@@ -442,7 +442,7 @@ $$
 
 The last equation is exactly the second equation of the second system.
 
-The crucial thing to note is that these operations can be undone: if in the second system the first equation is added three times to the second equation and added twice to the third equation we end up with the original system. So in fact we have
+The crucial thing to note is that these operations can be undone.  If in the second system the first equation is added three times to the second equation and added twice to the third equation we end up with the original system. So in fact we have
 
 $$
 
@@ -460,7 +460,7 @@ $$
    \right.
 $$
 
-The implication works two ways, which we can write as follows:
+The implication works two ways, which we can write as follows.
 
 $$
 
@@ -776,7 +776,7 @@ with $b\neq 0$ pops up, the system is inconsistent.
 </li>
 <li>
 
-If no such equation appears, the general solution can be found by back substitution: starting from the last equation, we work our way upwards.
+If no such equation appears, the general solution can be found by backward substitution: starting from the last equation, we work our way upwards.
 
 </li>
 </ul>
@@ -832,11 +832,10 @@ $$
 
 is a $3\times 5$ matrix.
 
-Its second row is $\begin{bmatrix} 2 & 7 & -1 & 0 & 8  \end{bmatrix}$ and its third column:
-
-$$
-\left[  \begin{array}{c} 3 \\ -1 \\ 5   \end{array}\right]
-$$
+Its second row is $\begin{bmatrix} 2 & 7 & -1 & 0 & 8  \end{bmatrix}$ and its third column is 
+$
+\left[  \begin{array}{c} 3 \\ -1 \\ 5   \end{array}\right].
+$
 
 ::::
 
@@ -965,15 +964,15 @@ $$
 \end{array}
 $$
 
-As we have seen before, the solution can now be found by back substitution.
+As we have seen before, the solution can now be found by backward substitution.
 
-The right moment to start this back substitution is when the augmented matrix has been simplified to so-called **echelon form**.
+The right moment to start this backward substitution is when the augmented matrix has been simplified to so-called **echelon form**.
 
 (Subsec:LinSystems:RowReduction)=
 
 ## Row Reduction and Echelon Forms
 
-In {numref}`Subsec:LinSystems:Elimination` we have solved linear systems by eliminating variables from the equations. It would be nice to have a clear mark where we can stop rewriting the given system, to forestall ending up in a never ending loop. When we use the notation of an augmented matrix we can identify such a mark.
+In {numref}`Subsec:LinSystems:Elimination` we have solved linear systems by eliminating variables from equations. It would be nice to have a clear mark where we can stop rewriting the given system, to forestall ending up in a never ending loop. When we use the notation of an augmented matrix we can identify such a mark.
 We first need a few more definitions.
 
 ::::{prf:definition}
@@ -1002,7 +1001,7 @@ Such a matrix is also called a **row echelon matrix**.
 :label: Ex:LinSystems:EchelonForm
 
 The following three matrices are meant to visualize the structure of an echelon matrix.
-The symbol $\blacksquare$ denotes an arbitrary nonzero number and $\ast$ just any real number.
+The symbol $\blacksquare$ denotes an arbitrary *nonzero* number and $\ast$ just any real number.
 
 $$
   E_1 =
@@ -1033,7 +1032,7 @@ However, since we will only consider row echelon matrices we will not do this. I
 ::::{prf:definition}
 :label: Dfn:LinSystems:Pivot
 
-A **pivot** of a row in an echelon matrix is the first nonzero element (the so-called _leading entry_) of the row.
+A **pivot** of a row in an echelon matrix is the first nonzero element (the so-called _leading entry_) of that row.
 
 ::::
 
@@ -1090,7 +1089,7 @@ Note that from the second condition in {prf:ref}`Dfn:LinSystems:EchelonForm` it
 
 ::::
 
-Now have a look again at the derivation at the end of the previous subsection. We worked our way downwards through the rows to create zeros in the first columns, while keeping in mind that we did not change the solution set of the corresponding linear system. The process is called **row reduction**. The end point, from which we could start building the solution by back substitution, was an augmented matrix in echelon form!
+Now have a look again at the derivation at the end of the previous subsection. We worked our way downwards through the rows to create zeros in the first columns, while keeping in mind that we did not change the solution set of the corresponding linear system. The process is called **row reduction**. The end point, from which we could start building the solution by backward substitution, was an augmented matrix in echelon form!
 
 ::::{prf:definition}
 :label: Dfn:LinSystems:RowOperations
@@ -1119,7 +1118,7 @@ Interchanging (or: swapping) two rows.
 
 ::::{prf:remark}
 
-Note that these row operations match exactly the operations of {prf:ref}`Prop:LinSystems:ElimOperations`. This proposition stated that the row operations do not change the solutions of the corresponding linear system.
+Note that these row operations match exactly the operations of {prf:ref}`Prop:LinSystems:ElimOperations`. This proposition now states that the row operations do not change the solutions of the corresponding linear system.
 ::::
 
 ::::{prf:definition}
@@ -1214,7 +1213,7 @@ $$
   \end{array}
 $$
 
-Here a row swap was essential to bring the matrix into echelon form. Sometimes a row swap may just be convenient to simplify the computations. Note that we have also introduced a notation for a row swap. It is good practice to use a notation like this when you do a row reduction process yourself.
+Here a row swap was essential to bring the matrix into echelon form. Sometimes a row swap may just be convenient to simplify the computations. Note that we have also introduced a notation for a row swap. We stress again that it is *good practice* to use a notation like this when you do a row reduction process yourself.
 To speed up the process it may be preferable to combine row operations that do not interfere. In this example the second and the third step both involved
 adding multiples of the first row to the other rows. This can be done simultaneously:
 
@@ -1287,7 +1286,7 @@ And so on, until we get to the last row, or until we get to a row below which al
 
 ::::
 
-The echelon matrix to which a matrix can be reduced is in no way unique. For instance, by scaling a row in an echelon matrix the echelon form persists. We can go a bit further, namely we can create zeros in the columns above the pivots as well. The following example shows how. First we work downwards to the echelon form and then work upwards to create the extra zeros, as mentioned.
+The echelon matrix to which a matrix can be reduced is in no way unique. For instance, by scaling a row in an echelon matrix the echelon form persists. We can go a bit further, namely we can create zeros in the columns *above* the pivots as well. The following example shows how. First we work downwards to the echelon form and then work upwards to create the extra zeros, as mentioned.
 
 ::::{prf:example}
 :label: Ex:LinSystems:RowReduction-2
@@ -1506,8 +1505,7 @@ and we add: '$x_3$ and $x_5$ are free'.
 
 ::::
 
-The row reduction of the augmented matrix to echelon form corresponds to the forward substitution as in {prf:ref}`Ex:LinSystems:SolWithBacksubst1` and {prf:ref}`Ex:LinSystems:SolWithBacksubst2`. There we found the solution by back-substitution. When the augmented matrix is reduced to reduced echelon form
-we have in fact incorporated this back-substitution part and can write down the general solution directly. We think that to solve a system 'with pen and paper', working with the reduced echelon matrix is less error prone. This holds in particular for a system where the solution contains one or more free variables.
+The row reduction of the augmented matrix to echelon form corresponds to the forward substitution as in {prf:ref}`Ex:LinSystems:SolWithBacksubst1` and {prf:ref}`Ex:LinSystems:SolWithBacksubst2`. There we found the solution by backward substitution. When the augmented matrix is reduced to reduced echelon form we have in fact incorporated this backward substitution part and can write down the general solution directly. We think that to solve a system 'with pen and paper', working with the reduced echelon matrix is less error prone. This holds in particular for a system where the solution contains one or more free variables.
 
 ::::{prf:theorem}
 :label: Thm:LinSystems:RowEquivalentToRREF
@@ -1520,11 +1518,11 @@ Any matrix is row equivalent to a reduced echelon matrix. Moreover, this last ma
 :label: Rem:LinSystems:RowEquivalentToRREF
 
 Again we give no formal proof.
-In the previous section we showed, also informally, that any matrix can be reduced to a matrix in echelon form.
+In the previous subsection we showed, also informally, that any matrix can be reduced to a matrix in echelon form.
 
 In this echelon matrix we may divide each row by its pivot (first nonzero element).
 
-And lastly working upwards' step by step we use a pivot -- which we made equal to 1 -- to create zeros in all positions above it.
+And lastly 'working upwards' step by step we use each pivot -- which we made equal to 1 -- to create zeros in all positions above it.
 
 (The last two simplifications may be done in reversed order: first use the pivots to create zeros in the positions above them and then scale the rows.)
 This reasoning supports the validity of the first statement.
@@ -1535,14 +1533,14 @@ The uniqueness is harder to show in an intuitive way, and it is definitely harde
 ::::{prf:example}
 :label: Ex:LinSystems:ReductionToRREF-1
 
-We further simplify the echelon matrix
+Let us further simplify the echelon matrix
 
 $$
 \begin{bmatrix} 3 & 2 &1 &6&-2\\   0 & 2 & -2 &-3 & 1\\  0 & 0 & 0 &3 & 2
    \end{bmatrix}
 $$
 
-to reduced echelon form:
+to reduced echelon form.
 
 step 1: use the pivot in the third row to create zeros above it;
 
@@ -1581,7 +1579,7 @@ $$
 
 ::::
 
-Instead of a formal proof of the uniqueness of the row reduced echelon form of a matrix, we illustrate this uniqueness with one example.
+Instead of a formal proof of the *uniqueness* of the row reduced echelon form of a matrix, we illustrate this uniqueness with one example.
 
 ::::{prf:example}
 :label: Ex:LinSystems:ReductionToRREF-2
@@ -1742,7 +1740,7 @@ If there is a pivot in the last column (the column 'behind the bar'), the system
 </li>
 <li>
 
-If the last column does not contain a pivot: write down the corresponding system of equations and express the variables in the pivot columns into the other variables (if any). These other variables are free variables.
+If the last column does not contain a pivot, write down the corresponding system of equations and express the variables in the pivot columns into the other variables (if any). These other variables are free variables.
 
 </li>
 </ol>
@@ -1759,10 +1757,10 @@ The word 'elimination' refers to the fact that the zeros that are created in the
 
 ::::
 
-::::{prf:remark} Row reduced echelon matrix versus back substitution
+::::{prf:remark} Row reduced echelon matrix versus back-substitution
 :label: Rem:LinSystems:RowRedVersusBackSubstitution
 
-We started this section by solving a linear system by reducing it to an equivalent system in echelon form and then use back-substitution. This is still a viable option. The advantage of the method described in {prf:ref}`Alg:LinSystems:ElimMethod` is that it avoids the clutter back substitution may lead to in the case of free variables.
+We started this section by solving a linear system by reducing it to an equivalent system in echelon form and then use back-substitution. This is still a viable option. The advantage of the method described in {prf:ref}`Alg:LinSystems:ElimMethod` is that it avoids the clutter back-substitution may lead to in the case of free variables.
 
 ::::
 
@@ -1771,7 +1769,7 @@ The following important general statement about the solution set of linear syste
 ::::{prf:theorem}
 :label: Thm:LinSystems:ZeroOneInfSolutions
 
-A system of linear equations has either zero, or one, or infinitely many solutions. In the case when there is exactly one solution, we speak of a **unique** solution.
+A system of linear equations has either zero, or one, or infinitely many solutions. In the case when there is exactly one solution we speak of a **unique** solution.
 
 ::::
 
@@ -1904,7 +1902,7 @@ $$
  a_1x_1 + a_2x_2 + a_3x_3 = b
 $$
 
-can be seen as a plane in $\mathbb{R}^3$. The previous proposition tells us: the intersection of $m$ planes in $\mathbb{R}^3$, where $m < 3$, cannot be a single point.
+can be seen as a plane in $\mathbb{R}^3$. The previous proposition tells us that the intersection of $m$ planes in $\mathbb{R}^3$, where $m < 3$, cannot be a single point.
 
 ::::
 
diff --git a/Chapter2/SolutionSets.md b/Chapter2/SolutionSets.md
index b64a3befa824eb2b04d452320bacf497e3a14385..9c047b13d0d0bfd51fce51f63dec17d5987ae8d1 100644
--- a/Chapter2/SolutionSets.md
+++ b/Chapter2/SolutionSets.md
@@ -144,12 +144,10 @@ What happens if we make the constant terms non-zero? Let us change the right han
 
 :::{math}
 
-$$
     \left\{\begin{array}{ccccccc}
     x_{1}&+&3x_{2}&-&x_{3}&=&a\\
     2x_{1}&+&6x_{2}&-&2x_{3}&=&2a
     \end{array}\right.
-$$
 
 :::
 
@@ -321,6 +319,7 @@ By putting $c_{2}=0$ in statement [ii.](#Item:SolSet:LinCombinSolSet), we find t
 By putting $c_{1}=1=c_{2}$, we find that any sum of two solutions is again a solution.
 
 </li>
+</ol>>
 
 You should keep in mind, however, that this **only** holds for solutions of **homogeneous** systems!
 
diff --git a/Chapter3/GeometryofLinearTransformations.md b/Chapter3/GeometryofLinearTransformations.md
index 21d74c9cb7b9e55017cfd38fd0306b5fe32ee46a..bec6e0a897507b7bb252705daa891f2f293d8d8f 100644
--- a/Chapter3/GeometryofLinearTransformations.md
+++ b/Chapter3/GeometryofLinearTransformations.md
@@ -146,11 +146,13 @@ $$
 
 this projection is depicted on the right in {numref}`Figure %s <Fig:GeomLinTrans:ProjinR2>`. It is an example of a non-orthogonal (or **oblique**) projection. Of course, we again have to check that this is really a linear transformation.
 
-::::{figure} Images/Fig-GeomLinTrans-ProjinR2.svg
+```{applet}
+:url: geom_lin_trans/proj_in_r2
+:fig: Images/Fig-GeomLinTrans-ProjinR2.svg
 :name: Fig:GeomLinTrans:ProjinR2
 
 On the left an orthogonal projection $T_{1}$ acting on a few selected vectors $\mathbf{u}_{1}$, $\mathbf{u}_{2}$, and $\mathbf{u}_{3}$. On the right a non-orthogonal projection $T_{2}$ acting on some selected vectors $\mathbf{v}_{1}$, $\mathbf{v}_{2}$, and $\mathbf{v}_{3}$. In both cases, the blue line represents the line $\mathcal{L}$ in the direction of $\begin{bmatrix}2\\1\end{bmatrix}$. On the left, every vector $\mathbf{u}_{i}$ is mapped to the closest vector that lies on $\mathcal{L}$. On the right, every vector $\mathbf{v}_{i}$ is mapped to the intersection of $\mathcal{L}$ wih the line through $\mathbf{v}_{i}$ in the direction given by $\begin{bmatrix}-2\\1\end{bmatrix}$.
-::::
+```
 
 ::::::{prf:proposition}
 Let $\mathcal{L}$ be a line through the origin and let $\mathbf{w}$ be a vector not on $\mathcal{L}$. The transformation $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ which maps a vector $\mathbf{u}$ to the intersection of $\mathcal{L}$ and the line through $\mathbf{u}$ in the direction of $\mathbf{w}$ is a linear transformation.
@@ -243,10 +245,10 @@ $$
 \mathbf{0}\quad\text{and in the direction of}\quad \mathbf{v}=\begin{bmatrix}1\\1\end{bmatrix}.
 $$
 
-We can define a transformation $T$ which reflects points in the plane along $\mathcal{L}$. (See {numref}`Figure %s <Fig:GeomLinTrans:ReflinR2>`.) It is easy to find the standard matrix $R$ of this transformation: the first standard basis vector $\mathbf{e}_{1}$ is mapped to $\mathbf{e}_{2}$ and, similarly, $\mathbf{e}_{2}$ is mapped to $\mathbf{e}_{1}$, so
+We can define a transformation $T$ which reflects points in the plane along $\mathcal{L}$. That is, we can imagine $\mathcal{L}$ to act as a kind of mirror, sending points on one side of $\mathcal{L}$ to their reflection on the other side. (See {numref}`Figure %s <Fig:GeomLinTrans:ReflinR2>`.) It is easy to find the standard matrix $M$ of this transformation: the first standard basis vector $\mathbf{e}_{1}$ is mapped to $\mathbf{e}_{2}$ and, similarly, $\mathbf{e}_{2}$ is mapped to $\mathbf{e}_{1}$, so
 
 $$
-R=\begin{bmatrix}0&1\\1&0\end{bmatrix}.
+M=\begin{bmatrix}0&1\\1&0\end{bmatrix}.
 $$
 
 ```{applet}
@@ -255,7 +257,7 @@ $$
 :name: Fig:GeomLinTrans:ReflinR2
 :status: reviewed
 
-The reflection $R$ along the line $\mathcal{L}$ in the direction of $\mathbf{v}=\begin{bmatrix}1\\1\end{bmatrix}$. The vectors in red are mapped to the vector in blue by this reflection.
+The reflection along the line $\mathcal{L}$ in the direction of $\mathbf{v}=\begin{bmatrix}1\\1\end{bmatrix}$. The vectors in red are mapped to the vector in blue by this reflection.
 ```
 
 So far so good. But how do we find the reflection over an arbitrary line $\mathcal{L}$? It turns out that the projections we have seen in Section {ref}`Subsec:GeomLinTrans:Proj` will help us out. Consider a line $\mathcal{L}$ and a vector $\mathbf{v}$ not in $\mathcal{L}$, as in {numref}`Figure %s <Fig:GeomLinTrans:ReflFromDoubleProj>`. In order to reflect $\mathbf{v}$ over $\mathcal{L}$, we first move it to the closest point on $\mathcal{L}$ and then move it the same distance again in the same direction.
@@ -300,15 +302,15 @@ A reflection is by definition a sum of scaled linear transformations. As such, i
 The following proposition guarantees that, as you would expect, applying a reflection twice leaves you back where you started.
 
 ::::::{prf:proposition}
-If $R$ is the standard matrix of a reflection, then $R^{2}=I$.
+If $M$ is the standard matrix of a reflection, then $R^{2}=I$.
 
 ::::::
 
 ::::::{prf:proof}
-We know that $R=2P-I$ where $P$ is the standard matrix of some projection. By {prf:ref}`Prop:GeomLinTrans:ProjSquaredisProj`, we have $P^{2}=P$ and therefore
+We know that $M=2P-I$ where $P$ is the standard matrix of some projection. By {prf:ref}`Prop:GeomLinTrans:ProjSquaredisProj`, we have $P^{2}=P$ and therefore
 
 $$
-R^{2}=(2P-I)(2P-I)=4P^{2}-4P+I=I.
+M^{2}=(2P-I)(2P-I)=4P^{2}-4P+I=I.
 $$
 
 ::::::
@@ -321,7 +323,7 @@ The definition of a reflection in combination with {prf:ref}`Prop:GeomLinTrans:M
 Let $\mathcal{L}$ be the line in the plane that passes through the origin and that makes an angle $\theta$ with the positive $x$-axis. The standard matrix of the reflection along $\mathcal{L}$ is
 
 $$
-A_{R_{\mathcal{L}}}=2\begin{bmatrix}\cos^{2}(\theta)&\sin(\theta)\cos(\theta)\\\sin(\theta)\cos(\theta)&\sin^{2}(\theta)\end{bmatrix}-I_{2}=\begin{bmatrix}\cos(2\theta)&\sin(2\theta)\\\sin(2\theta)&-\cos(2\theta)\end{bmatrix}.
+M_{\mathcal{L}}=2\begin{bmatrix}\cos^{2}(\theta)&\sin(\theta)\cos(\theta)\\\sin(\theta)\cos(\theta)&\sin^{2}(\theta)\end{bmatrix}-I_{2}=\begin{bmatrix}\cos(2\theta)&\sin(2\theta)\\\sin(2\theta)&-\cos(2\theta)\end{bmatrix}.
 $$
 
 ::::::
@@ -353,7 +355,7 @@ One particularly interesting aspect of reflections is that they preserve lengths
 ::::{prf:proposition}
 :label: Prop:GeomLinTrans:ReflDotProd
 
-If $S$ is a defined on $\R^{n}$, then for any $\vect{w}_{1},\vect{w}_{2}$ in $\R^{n}$ we have:
+If $S$ is a reflection defined on $\R^{n}$, then for any $\vect{w}_{1},\vect{w}_{2}$ in $\R^{n}$ we have:
 
 $$S(\vect{w}_{1})\cdot S(\vect{w}_{2})=\vect{w}_{1}\cdot\vect{w}_{2}.$$
 
@@ -364,7 +366,7 @@ $$S(\vect{w}_{1})\cdot S(\vect{w}_{2})=\vect{w}_{1}\cdot\vect{w}_{2}.$$
 By definition, there is an orthogonal projection with standard matrix $P$ such that $S(\vect{w})=(2P-I)\vect{w}$. We assume $P$ is the projection on the span of a single vector $\vect{v}$. If there are more, the computations become considerably messier, but neither harder nor more enlightening.
 
 \begin{align*}
-S(\vect{w}*{1})\cdot S(\vect{w}_{2})&=(2P-I)\vect{w}_{1}\cdot(2P-I)\vect{w}_{2}\\
+S(\vect{w}_{1})\cdot S(\vect{w}_{2})&=(2P-I)\vect{w}_{1}\cdot(2P-I)\vect{w}_{2}\\
 &=(2\left(\frac{\vect{w_{1}}\cdot\vect{v}}{\vect{v}\cdot\vect{v}}\right)\vect{v}-\vect{w}_{1})\cdot (2\left(\frac{\vect{w_{2}}\cdot\vect{v}}{\vect{v}\cdot\vect{v}}\right)\vect{v}-\vect{w}_{2})\\
 &=4\left(\frac{\vect{w}_{1}\cdot \vect{v}}{\vect{v}\cdot\vect{v}}\right)\left(\frac{\vect{w}_{2}\cdot \vect{v}}{\vect{v}\cdot\vect{v}}\right)\vect{v}\cdot\vect{v}-2\left(\frac{\vect{w}_{2}\cdot \vect{v}}{\vect{v}\cdot\vect{v}}\right)\vect{w}_{1}\cdot\vect{v}-2\left(\frac{\vect{w}_{1}\cdot \vect{v}}{\vect{v}\cdot\vect{v}}\right)\vect{w}_{2}\cdot\vect{v}+\vect{w}_{1}\cdot\vect{w}_{2}\\
 &=\vect{w}_{1}\cdot\vect{w}_{2}
@@ -457,13 +459,13 @@ Any rotation in the plane is the composition of two reflections.
 ::::::
 
 ::::::{prf:proof}
-We will show that the standard matrix $R_{\theta}$ of the rotation over an angle $\theta$ is the product of the standard matrices of two reflections. The claim follows then from the definition of the matrix product.
+We will show that the standard matrix $R_{\theta}$ of the rotation $T_{\theta}$ over an angle $\theta$ is the product of the standard matrices of two reflections. The claim follows then from the definition of the matrix product.
 
-Let $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ be two lines in the plane through the origin that make an angle of $\theta/2$ with each other. If we call $\phi/2$ the angle $\mathcal{L}_{1}$ makes with the positive $x$-axis, we can conclude that $\mathcal{L}_{2}$ makes an angle of $\phi/2+\theta/2$ with the positive $x$-axis. From {prf:ref}`Prop:GeomLinTrans:MatofReflinPlane`, we know that the standard matrices of the reflections along $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are
+Let $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ be two lines in the plane through the origin that make an angle of $\theta/2$ with each other. If we call $\phi/2$ the angle $\mathcal{L}_{1}$ makes with the positive $x$-axis, we can conclude that $\mathcal{L}_{2}$ makes an angle of $\phi/2+\theta/2$ with the positive $x$-axis. From {prf:ref}`Prop:GeomLinTrans:MatofReflinPlane`, we know that the standard matrices of the reflections $T_{\mathcal{L}_{1}}$ and $T_{\mathcal{L}_{2}}$ along $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$, respectively, are
 
 $$
-R_{\mathcal{L}_{1}}=\begin{bmatrix}\cos(\phi)&\sin(\phi)\\\sin(\phi)&-\cos(\phi)\end{bmatrix}\quad
-\text{and}\quad R_{\mathcal{L}_{2}}=\begin{bmatrix}\cos(\theta+\phi)&\sin(\theta+\phi)\\
+M_{\mathcal{L}_{1}}=\begin{bmatrix}\cos(\phi)&\sin(\phi)\\\sin(\phi)&-\cos(\phi)\end{bmatrix}\quad
+\text{and}\quad M_{\mathcal{L}_{2}}=\begin{bmatrix}\cos(\theta+\phi)&\sin(\theta+\phi)\\
 \sin(\theta+\phi)&-\cos(\theta+\phi)\end{bmatrix},
 
 
@@ -477,7 +479,7 @@ respectively. Using the fact that, for any angles $\alpha$ and $\beta$, we have
 we find
 
 $$
-R_{\mathcal{L}_{2}}R_{\mathcal{L}_{1}}=\begin{bmatrix}
+M_{\mathcal{L}_{2}}M_{\mathcal{L}_{1}}=\begin{bmatrix}
 \cos(\theta)&\sin(-\theta)\\\sin(\theta)&\cos(\theta)
 \end{bmatrix}=R_{\theta}.
 
@@ -486,11 +488,13 @@ $$
 
 ::::::
 
-::::{figure} Images/Fig-GeomLinTrans-RotisDoubleRefl.svg
+```{applet}
+:url: geom_lin_trans/rotisdoublerefl
+:fig: Images/Fig-GeomLinTrans-RotisDoubleRefl.svg
 :name: Fig:GeomLinTrans:RotisDoubleRefl
 
 {prf:ref}`Prop:GeomLinTrans:RotisDoubleRefl` illustrated. $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are arbitrary lines that make an angle of $\theta/2$ with each other. Composing the reflections along $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ then gives the rotation over the angle $\theta$. This is shown for the particular vector $\mathbf{v}$. Note that the angle $\phi/2$ that $\mathcal{L}_{1}$ makes with the positive $x$ axis is irrelevant to the proof.
-::::
+```
 
 In the plane, you can only rotate around the origin. Things get considerably more complicated if we move to $\mathbb{R}^{3}$, because there you can rotate around any arbitrary line. We will not get into that here.
 
@@ -580,8 +584,8 @@ If $T$ is a shear transformation fixing the line $\mathcal{L}$ and $\mathbf{v}$
 ::::::{prf:proof}
 The distance between a vector $\mathbf{v}$ and a line $\mathcal{L}$ is the length of $\mathbf{v}-\proj_{\mathcal{L}}(\mathbf{v})$. We find, for arbitrary $\mathbf{v}$ in $\mathbb{R}^{2}$,
 \begin{align*}
-\lVert T(\mathbf{v})-\proj*{\mathcal{L}}(T(\mathbf{v}))\rVert &= \lVert \mathbf{v}+c\mathbf{w}-\frac{(\mathbf{v}+c\mathbf{w})\ip\mathbf{w}}{\mathbf{w}\ip\mathbf{w}}\mathbf{w}\rVert=\lVert \mathbf{v}-\frac{\mathbf{v}\ip\mathbf{w}}{\mathbf{w}\ip\mathbf{w}}\mathbf{w}\rVert\\
-&=\lVert \mathbf{v} -\proj*{\mathcal{L}}\mathbf{v}\rVert
+\lVert T(\mathbf{v})-\proj_{\mathcal{L}}(T(\mathbf{v}))\rVert &= \lVert \mathbf{v}+c\mathbf{w}-\frac{(\mathbf{v}+c\mathbf{w})\ip\mathbf{w}}{\mathbf{w}\ip\mathbf{w}}\mathbf{w}\rVert=\lVert \mathbf{v}-\frac{\mathbf{v}\ip\mathbf{w}}{\mathbf{w}\ip\mathbf{w}}\mathbf{w}\rVert\\
+&=\lVert \mathbf{v} -\proj_{\mathcal{L}}\mathbf{v}\rVert
 \end{align*}
 which had to be proven.
 
@@ -615,7 +619,7 @@ $$
 
 ::::::{prf:example} Application
 
-Suppose we have a standard deck of $52$ playing cards placed in a stack on a table. A standard playing card is about $87$ by $56$ millimeters, so we can assume that the corners of the lowest card are on
+Suppose we have a standard deck of $52$ perfectly rectangular playing cards placed in a stack on a table. A standard playing card is about $87$ by $56$ millimeters, so we can assume that the corners of the lowest card are on
 
 $$
 \begin{bmatrix}0\\0\\0\end{bmatrix},\quad\begin{bmatrix}87\\0\\0\end{bmatrix},\quad\begin{bmatrix}0\\56\\0\end{bmatrix},\quad\text{and}\quad\begin{bmatrix}87\\56\\0\end{bmatrix},
diff --git a/Chapter4/BasisAndDimension.md b/Chapter4/BasisAndDimension.md
index ccbda1f1a509d3c0d7b24a86c8e6f3614354df77..a36cec887fea396ab1bb7d9f40713c0e2bbbfcf3 100644
--- a/Chapter4/BasisAndDimension.md
+++ b/Chapter4/BasisAndDimension.md
@@ -159,11 +159,13 @@ This basis was already defined in {numref}`Sec:LinTrafo`, Equation {eq}`Eq:LinTr
 The standard basis in $\R^2$ can be visualized as two perpendicular unit vectors, e.g.
 with $\vect{e}_1$ pointing to the right and $\vect{e}_2$ pointing upwards. Likewise we can depict the standard basis $\mathcal E$ of $\R^3$. See {numref}`Figure %s <Fig:BasisDim:StandardBasis>`.
 
-::::{figure} Images/Fig-BasisDim-StandardBasis.svg
+```{applet}
+:url: basisdim/standardbasis
+:fig: Images/Fig-BasisDim-StandardBasis.svg
 :name: Fig:BasisDim:StandardBasis
 
 The standard bases in $\R^2$ and $\R^3$
-::::
+```
 
 ::::::
 
@@ -991,7 +993,7 @@ c_2
 = c_1\vect{u}_1 + c_2\vect{u}_2.
 $$
 
-Thus $\{\vect{u}_1, \vect{u}_2\}$  is a basis for $\mathcal{P}$.
+Thus $\{\vect{u}_1, \vect{u}_2\}$ is a basis for $\mathcal{P}$.
 
 We can also take $x_1$ and $x_3$ as free variables, and then find
 
@@ -1173,7 +1175,7 @@ Invoking {prf:ref}`Thm:LinInd:TooManyVectsimpliesLinDep` we get that
 $\lbrace\vect{b}_1, \vect{b}_2, \ldots, \vect{b}_k, \vect{s} \rbrace$
 is linearly dependent.
 
-Since we assumed $ \lbrace\vect{b}_1, \vect{b}_2, \ldots, \vect{b}_k\rbrace$ is linearly independent, we may conclude that
+Since we assumed $\lbrace\vect{b}_1, \vect{b}_2, \ldots, \vect{b}_k\rbrace$ is linearly independent, we may conclude that
 
 $$
 \vect{s} \in \Span{\vect{b}_1, \vect{b}_2, \ldots, \vect{b}_k}.
@@ -1438,7 +1440,7 @@ $$
 \Row{B} \subseteq  \Row{A}.
 $$
 
-Since the equivalence works  both ways we can interchange $A$ and $B$ and conclude that also
+Since the equivalence works both ways we can interchange $A$ and $B$ and conclude that also
 
 $$
 \Row{A}  \subseteq  \Row{B},
diff --git a/Chapter4/ChangeOfBasis.md b/Chapter4/ChangeOfBasis.md
index 3af4131a919a2f9eaa7694089bd00e0a6a2e928b..cdcf61674641aae6ab40536a67a9568597f951da 100644
--- a/Chapter4/ChangeOfBasis.md
+++ b/Chapter4/ChangeOfBasis.md
@@ -63,7 +63,7 @@ We start with a proposition.
 
 If $\mc{B} = \{\vect{b}_1, \vect{b}_2,\ldots,\vect{b}_m\}$ is a basis for a subspace $S$ in $\R^n$,
 then any vector $\vect{v}$ in $S$ can written as a linear combination of
-$\vect{b}_1, \vect{b}_2,\ldots,\vect{b}_m$ in a **unique** way, i.e.
+$\vect{b}_1, \vect{b}_2,\ldots,\vect{b}_m$ in a *unique* way, i.e.
 
 $$
   \vect{v} = c_1\vect{b}_1 + c_2\vect{b}_2 + \ldots + c_m\vect{b}_m,
@@ -82,7 +82,7 @@ $$
               \{\vect{b}_1, \ldots, \vect{b}_m\} \text{ linearly independent}.
 $$
 
-Let $ \vect{v}$ be an arbitrary element of $S$. As $ \vect{v} \in \Span{\vect{b}_1, \ldots, \vect{b}_m}$,
+Let $ \vect{v}$ be an arbitrary element of $S$. As $\vect{v} \in \Span{\vect{b}_1, \ldots, \vect{b}_m}$,
 
 $$
    \vect{v} = c_1\vect{b}_1 + c_2\vect{b}_2 + \ldots + c_m\vect{b}_m
@@ -174,7 +174,7 @@ $$
 $$
 
 Likewise the vector $\vect{w}$ has the coordinate vector
-$ [\vect{w}]_{\mc B} = \begin{bmatrix} 2 \\ -2\end{bmatrix}$.
+$[\vect{w}]_{\mc B} = \begin{bmatrix} 2 \\ -2\end{bmatrix}$.
 
 Namely,
 
@@ -346,12 +346,11 @@ $$
 Show that every change-of-coordinates matrix $P_{\mc{B}}$ is invertible.
 ::::
 
-
 ::::{dropdown} Solution to&nbsp;{numref}`Exc:ChangeOfBasis:InvertiblePB`&nbsp;(_click to show_)
 
 Let $\mc{B} = \{\vect{b}_1,\vect{b}_2, \ldots, \vect{b}_n\}$ be any basis of $\R^n$.
 <BR>
-By definition 
+By definition
 
 $$
    P_{\mc{B}} = [\,\vect{b}_1\,\,\vect{b}_2\,\,\ldots\,\,\vect{b}_n\,].
@@ -479,7 +478,6 @@ Because of the linearity of $T$ we then have
 :::{math}
 :label: Eq:ChangeOfBasis:LinearityTransform
 
-
 T(\vect{v}) = f_1T(\vect{b}_1) + f_2T(\vect{b}_2) + \ldots + f_nT(\vect{b}_n).
 
 :::
@@ -623,34 +621,43 @@ $$
 
 ::::
 
-::::{exercise}
-:label: Exc:ChangeOfBasis:FindSimpleMatrix
+::::{grasple} 
+:url: https://embed.grasple.com/exercises/d6aa929c-7614-4af2-b015-31e2e6fd7b80?id=104847
+:label: grasple_exercise_4_3_T1
+:dropdown:
+:description: Finding  $[T]_{\mathcal{B}}$ for a 'nice' linear transformation.
 
-Consider the linear transformation $T$ with the standard matrix $A$ given by
+::::
 
-$$
-  T:\R^3 \to \R^3, \quad T(\vect{x}) = A\vect{x} =
-  \begin{bmatrix} -2 & 1 & 0 \\ -7 & 3 & 1 \\ 2 & 0 & -1
-  \end{bmatrix}\vect{x}.
-$$
 
-It is given that
 
-$$
-  A\begin{bmatrix} 0 \\0\\1  \end{bmatrix} =\begin{bmatrix} 0 \\1\\-1  \end{bmatrix},
-  \quad
-  A\begin{bmatrix} 0\\1\\-1  \end{bmatrix} =\begin{bmatrix} 1  \\2 \\ 1  \end{bmatrix},
-  \quad \text{and } \quad
-  A\begin{bmatrix} 1 \\2\\1  \end{bmatrix} =\begin{bmatrix} 0\\0\\ 1  \end{bmatrix}.
-$$
+%::::{exercise}
+%:label: Exc:ChangeOfBasis:FindSimpleMatrix
 
-Find the matrix $[T]_{\mc{B}}$ for the basis
+%Consider the linear transformation $T$ with the standard matrix $A$ given by
 
-$$
-  \mc{B} = \left\{ \begin{bmatrix} 0 \\0\\ 1  \end{bmatrix},\,\begin{bmatrix} 0 \\1\\ -1  \end{bmatrix}, \,\begin{bmatrix} 1 \\2\\1  \end{bmatrix} \right\}.
-$$
+%$$
+%  T:\R^3 \to \R^3, \quad T(\vect{x}) = A\vect{x} =
+%  \begin{bmatrix} -2 & 1 & 0 \\ -7 & 3 & 1 \\ 2 & 0 & -1
+%  \end{bmatrix}\vect{x}.
+%$$
 
-::::
+%It is given that
+
+%$$
+%  A\begin{bmatrix} 0 \\0\\1  \end{bmatrix} =\begin{bmatrix} 0 \\1\\-1  \end{bmatrix},
+% A\begin{bmatrix} 0\\1\\-1  \end{bmatrix} =\begin{bmatrix} 1  \\2 \\ 1  \end{bmatrix},
+%  \quad \text{and } \quad
+%  A\begin{bmatrix} 1 \\2\\1  \end{bmatrix} =\begin{bmatrix} 0\\0\\ 1  \end{bmatrix}.
+%$$
+
+%Find the matrix $[T]_{\mc{B}}$ for the basis
+
+%$$
+%  \mc{B} = \left\{ \begin{bmatrix} 0 \\0\\ 1  \end{bmatrix},\,\begin{bmatrix} 0 \\1\\ -1  \end{bmatrix}, \,\begin{bmatrix} 1 \\2\\1  \end{bmatrix} \right\}.
+%$$
+
+%::::
 
 In the next example we will find the matrix of the rotation $R$ about the origin through an angle $\frac13\pi$. In {numref}`Sec:GeomLinTrans` we have already seen the standard matrix of this linear transformation, involving sines and cosines. As the example shows a change of basis may lead to an interesting alternative.
 
@@ -663,7 +670,7 @@ $$
   \left\{\vect{b}_1, \vect{b}_2 \right\} = \left\{ \begin{bmatrix} 2 \\ 0  \end{bmatrix}, \begin{bmatrix} -1 \\ \sqrt{3}  \end{bmatrix} \right\}.
 $$
 
-Note that $\vect{b}_1$ and $ \vect{b}\_2$ are two vectors of length 2 subtending an angle
+Note that $\vect{b}_1$ and $ \vect{b}_2$ are two vectors of length 2 subtending an angle
 $\frac23\pi$. See {numref}`Figure %s <Fig:ChangeOfBasis-TriangularGrid>`.
 The vector $\vect{b}_1+\vect{b}_2 = \begin{bmatrix} 1 \\ \sqrt{3}  \end{bmatrix}$, is also indicated in the figure.
 
@@ -695,9 +702,9 @@ In {prf:ref}`Ex:ChangeOfBasis:Reflection` and {prf:ref}`Ex:ChangeOfBasis:Rotatio
 
 (Subsec:ChangeOfBasis:RelationTETB)=
 
-## The Relation between $[T]_{\mc{E}}$ and $[T]_{\mc{B}}$
+## The relation between $[T]_{\mc{E}}$ and $[T]_{\mc{B}}$
 
-We have seen how to convert vectors from one coordinate system (i.e., basis) to another, and also how to construct the matrix of a linear transformation with respect to an arbitrary basis. In this section we will present a ready-made formula that connects the matrices with respect to two different bases. In this subsection we will restrict ourselves to the common situation of a linear transformation from $\R^n$ to itself,  where one of the bases is the standard basis.
+We have seen how to convert vectors from one coordinate system (i.e., basis) to another, and also how to construct the matrix of a linear transformation with respect to an arbitrary basis. In this section we will present a ready-made formula that connects the matrices with respect to two different bases. In this subsection we will restrict ourselves to the common situation of a linear transformation from $\R^n$ to itself, where one of the bases is the standard basis.
 
 We start with an example that illustrates the underlying ideas.
 
@@ -865,16 +872,14 @@ Suppose $P$ is an invertible matrix.
 Show that for matrices $A$ and $B$
 
 $$
-   B = P^{-1}\,A\,P \,\, \iff \,\,A = P\,B\,P^{-1}.
+   B = P^{-1}A\,P \,\, \iff \,\,A = P\,B\,P^{-1}.
 $$
 
 ::::
 
-
-
 ::::{dropdown} Solution to&nbsp;{numref}`Exc:ChangeOfBasis:PinvAPversusPBinvP`&nbsp;(_click to show_)
 
-Multiply both sides from the left by $P$ and from the right by $P^{-1}$,
+Starting from $B = P^{-1}A\,P$   multiply both sides from the left by $P$ and from the right by $P^{-1}$,
 and use that $P^{-1}P = PP^{-1} = I$:
 
 $$
@@ -883,7 +888,6 @@ $$
 
 ::::
 
-
 ::::{prf:example}
 :label: Ex:ChangeOfBasis:MatrixOrthProjection
 
@@ -896,12 +900,13 @@ $$
 A suitable basis here is a basis $\mc{B} = \{\vect{b}_1, \vect{b}_2,\vect{b}_3\}$ where the first two vectors lie in the plane, and the third vector is perpendicular to the plane.
 See {numref}`Figure %s <Fig:ChangeOfBasis:Projection>`.
 
-:::{figure} Images/Fig-ChangeOfBasis-Projection.svg
+```{applet}
+:url: change_of_basis/projection
+:fig: Images/Fig-ChangeOfBasis-Projection.svg
 :name: Fig:ChangeOfBasis:Projection
 
 Projection with respect to a suitable basis.
-
-:::
+```
 
 For instance, we can take
 
@@ -937,9 +942,10 @@ and we can write down the standard matrix in one stroke:
 
 This can be evaluated to yield 
 
-$$
+:::{math}
+:label: Eq:ChangeOfBasis:ProjMatrix2
   [T]_{\mc{E}} = \frac16 \begin{bmatrix}5 & 1 & 2\\1& 5& -2\\2&-2&2 \end{bmatrix}.
-$$
+:::
 
 ::::
 
@@ -949,15 +955,15 @@ $$
 As was shown in {prf:ref}`Prop:GeomLinTrans:ProjSquaredisProj` of {numref}`Subsec:GeomLinTrans:Proj` the matrix $A$ of a projection has to satisfy the identity
 $A^2=A$. Show that the matrix $[T]_{\mc{E}}$ of {prf:ref}`Ex:ChangeOfBasis:MatrixOrthProjection` indeed satisfies this identity.
 
-(To verify this you don't have to compute the matrix as given by Equation {eq}`Eq:ChangeOfBasis:ProjMatrix` explicitly)
+(To verify this you don't have to compute the matrix as given by Equation {eq}`Eq:ChangeOfBasis:ProjMatrix2`.)
 
 ::::
 
 ::::{dropdown} Solution to&nbsp;{numref}`Exc:ChangeOfBasis:CheckIdempotent`&nbsp;(_click to show_)
 
-Obviously the matrix  $B = [T]_{\mc{B}} =  \begin{bmatrix}1 & 0 & 0\\0& 1& 0\\0&0&0  \end{bmatrix}$   has the property  $B^2 = B$.
+Obviously the matrix $B = [T]_{\mc{B}} =  \begin{bmatrix}1 & 0 & 0\\0& 1& 0\\0&0&0  \end{bmatrix}$ has the property $B^2 = B$.
 
-It follows that the matrix  $A = P_{\mathcal{B}} B P_{\mathcal{B}}^{-1} = PBP^{-1}$ has it too:
+It follows that the matrix $A = P_{\mathcal{B}} B P_{\mathcal{B}}^{-1} = PBP^{-1}$ has it too:
 
 $$
    A^2 = PBP^{-1}PBP^{-1} = PB^2 P^{-1} = PBP^{-1} = A. 
@@ -968,7 +974,6 @@ $$
 In the last subsection we will generalize the formula to the more general situation of a linear transformation from $\R^n$ to $\R^m$ and arbitrary bases $\mc{B},\mc{B}'$ for $\R^n$, and bases
 $\mc{C},\mc{C}'$ for $\R^m$.
 
-
 (Subsec:ChangeOfBasis:GeneralTransformationFormula)=
 
 ## General Transformation Formula for Matrices of Linear Transformation
@@ -995,7 +1000,7 @@ $$
 
 ::::
 
-Again, as the name suggests, the change-of-basis matrix can be used to switch from coordinates with respect to one basis to coordinates with respect to another basis.
+Again, as the name suggests, the change-of-coordinates matrix can be used to switch from coordinates with respect to one basis to coordinates with respect to another basis.
 
 ::::{prf:proposition}
 :label: Prop:ChangeOfBasis:CoBmatrix
@@ -1033,13 +1038,14 @@ The last expression is a linear combination of $n$ vectors in $\R^n$ and consequ
 
 $$
    [\vect{v}]_{\mc C} = \big[ \,[\vect{b}_1]_{\mc C}\,\, \ldots \,\, [\vect{b}_n]_{\mc C} \,\big] \begin{bmatrix} p_1 \\ \vdots \\ p_n \end{bmatrix} =
-        \CoBmatrix{B}{C}[\vect{v}]_{\mc B}.
+        \CoBmatrix{B}{C}[\vect{v}]_{\mc B},
 $$
 
+as this is exactly how $\CoBmatrix{B}{C}[\vect{v}]$ was defined in {prf:ref}`Dfn:ChangeOfBasis:CoBmatrix`.
 ::::
 
 ::::{exercise}
-:label: Exc:ChangeOfBasis:RelationMatricesCoB
+:label: Exc:ChangeOfBasis:InverseOfMatrixCoB
 
 Suppose $\mc{B}$ and $\mc{C}$ are two bases of $\R^n$.
 Show that the change-of-coordinates matrix $P_{\mc{C}\leftarrow\mc{B}}$ is invertible, and that
@@ -1051,6 +1057,7 @@ $$
 ::::
 
 ::::{exercise}
+:label: Exc:ChangeOfBasis:RelationMatricesCoB
 
 Suppose $\mc{B}, \mc{C}$ and $\mc{D}$ are three bases of $\R^n$.
 What is the relation between the
@@ -1202,7 +1209,6 @@ must be equal.
 
 ::::
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/18d96cbf-6158-4800-9a10-3ec7f6e933f8?id=90885
 :label: grasple_exercise_4_3_3
@@ -1211,7 +1217,6 @@ must be equal.
 
 ::::
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/dfad4903-b2a8-4f67-979e-5651cf4072ec?id=90872
 :label: grasple_exercise_4_3_4
@@ -1220,7 +1225,6 @@ must be equal.
 
 ::::
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/d6e53d94-578e-48fe-96b6-aca26f4eca1c?id=90870
 :label: grasple_exercise_4_3_5
@@ -1229,7 +1233,6 @@ must be equal.
 
 ::::
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/75c6d18b-8a54-4592-97f4-edd77169cc10?id=90876
 :label: grasple_exercise_4_3_6
@@ -1238,85 +1241,86 @@ must be equal.
 
 ::::
 
+::::{grasple}
+:url: https://embed.grasple.com/exercises/c45ea239-1e90-4d1f-ae08-323e595bd53a?id=104827
+:label: grasple_exercise_4_3_7   
+:dropdown:
+:description: Expressing a vector $\vect{v} \in \R^3$  in  a basis $\mathcal{B} = \{\vect{b}_1,\vect{b}_2,\vect{b}_3\}$  (and vice versa).
+
+::::
+
+
+
 
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/39e1af19-0a32-456b-a414-20056e6b7f16?id=85157
-:label: grasple_exercise_4_3_7
+:label: grasple_exercise_4_3_8
 :dropdown:
 :description: Expressing a vector $\vect{v} \in \R^3$  in  a basis $\mathcal{B} = \{\vect{b}_1,\vect{b}_2,\vect{b}_3\}$.
 
 ::::
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/b03d9983-3ef2-4d45-82b3-6c1762510561?id=90875
-:label: grasple_exercise_4_3_8
+:label: grasple_exercise_4_3_9
 :dropdown:
 :description: To explain the relation between $\vect{x}$  and $[\vect{x}]_{\mathcal{B}}$.
 
 ::::
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/784bba91-a0be-4076-9918-63b8ab2fbc49?id=90881
-:label: grasple_exercise_4_3_9
+:label: grasple_exercise_4_3_10
 :dropdown:
 :description: To find out how coordinates change when a basis is reordered.
 ::::
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/d8894445-4426-4694-8500-229cd47a5288?id=85165
-:label: grasple_exercise_4_3_10
+:label: grasple_exercise_4_3_11
 :dropdown:
 :description: To commute between  $[\vect{v}]_{\mathcal{B}}$  and $[\vect{v}]_{\mathcal{C}}$  in $\R^2$.  
 
-:::: 
-
+::::
 
 The remaining exercises are about matrix representations of linear transformations.
 
-
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/2de2a3b5-1d3f-4e79-9421-393d59b9dc87?id=93047
-:label: grasple_exercise_4_3_11
+:label: grasple_exercise_4_3_12
 :dropdown:
 :description:  To transform from $[T]_{\mathcal{E}}$  to  $[T]_{\mathcal{B}}$  for $T$ from $\R^2$ to $\R^2$.
 
-:::: 
-
+::::
 
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/01e5c371-0139-458c-8f0a-25f35bc03fcb?id=93053
-:label: grasple_exercise_4_3_12
+:label: grasple_exercise_4_3_13
 :dropdown:
 :description: To transform from $[T]_{\mathcal{E}}$  to  $[T]_{\mathcal{B}_2\leftarrow\mathcal{B}_1}$ for $T$ from $\R^2$ to $\R^3$.
 
-:::: 
-
+::::
 
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/dd1af96d-34d7-407e-9877-c1e8b6495e6f?id=85167
-:label: grasple_exercise_4_3_13
+:label: grasple_exercise_4_3_14
 :dropdown:
 :description: To compute $[T]_{\mathcal{C}\leftarrow\mathcal{B}}$  for $T$ from $\R^2$ to $\R^2$  with respect to several bases. 
 
-:::: 
-
+::::
 
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/933f3e07-36b6-4db5-a47d-91e276185269?id=85159
-:label: grasple_exercise_4_3_14
+:label: grasple_exercise_4_3_15
 :dropdown:
 :description: To compute $[T]_{\mathcal{C}\leftarrow\mathcal{B}}$, for $T$ from $\R^2$ to $\R^2$.  with respect to several bases. 
 
-:::: 
-
+::::
 
 ::::{grasple}
 :url: https://embed.grasple.com/exercises/7985ac3f-c432-4b48-9ef1-8ce0914b0f97?id=85162
-:label: grasple_exercise_4_3_15
+:label: grasple_exercise_4_3_16
 :dropdown:
-:description: Like the previous,  for a linear transformation $T$ from $\R^3$ to $\R^2$.
+:description: Like the previous, for a linear transformation $T$ from $\R^3$ to $\R^2$.
 
-:::: 
\ No newline at end of file
+::::
diff --git a/Chapter5/DeterminantsExtras.md b/Chapter5/DeterminantsExtras.md
index 1d8a73bae51d6952af53e20a39e26f49ada14869..25a6fb34c34b6eb5d5c3975517866470bf08ea9a 100644
--- a/Chapter5/DeterminantsExtras.md
+++ b/Chapter5/DeterminantsExtras.md
@@ -441,7 +441,7 @@ By the linearity property (in all of the columns) of the determinant ({prf:ref}`
 :::{math}
 :label: Eq:DetExtras:ProofCramer
 
-c_1\det{(A)} + c_2\det{(A^{(1)}(\vect{a}\_2))} + \ldots + c_n\det{(A^{(1)}(\vect{a}\_n))} - \det{(A^{(1)}(\vect{b}))} = 0.
+c_1\det{(A)} + c_2\det{(A^{(1)}(\vect{a}_2))} + \ldots + c_n\det{(A^{(1)}(\vect{a}_n))} - \det{(A^{(1)}(\vect{b}))} = 0.
 
 :::
 
@@ -603,6 +603,7 @@ provided that $A$ is invertible.
 In fact a slightly more general formula holds for **any** square matrix.
 
 ::::::{prf:proposition}
+:label: Prop:DetExtras:AdjointProperty
 
 For any square matrix $A$ the following identity holds:
 
@@ -616,7 +617,7 @@ For clarity we used dots to indicate products. Note that the first two products
 
 The proof we think, is short and instructive.
 
-::::::{prf:proof}
+::::::{dropdown} Proof of&nbsp;{prf:ref}`Prop:DetExtras:AdjointProperty`
 
 For an invertible matrix the statement follows immediately from {prf:ref}`Prop:DetExtras:Inverse`.
 
@@ -896,7 +897,7 @@ so property iv. is satisfied too.
 
 We end the chapter with a proof of {prf:ref}`Prop:DetExtras:Properties-ndimCrossProd`.
 <BR>
-So, if you are interested,  push the  $\vee$  button on the right.
+So, if you are interested,  push the button on the right.
 
 ::::::{dropdown} Proof of&nbsp;{prf:ref}`Prop:DetExtras:Properties-ndimCrossProd`
 
diff --git a/Chapter5/DeterminantsGeometric.md b/Chapter5/DeterminantsGeometric.md
index 85e6cb6af916df12cbd73fe82f8fb4a6f4c7cf7f..236c2ad9d45c83ed27629622126408bf17c4a863 100644
--- a/Chapter5/DeterminantsGeometric.md
+++ b/Chapter5/DeterminantsGeometric.md
@@ -4,7 +4,7 @@
 
 ## Introduction
 
-The word "determinant" already appeared in {numref}`Section %s <Sec:CrossProduct>` about the cross product. And in {numref}`Subsection %s <Subsec:MatrixInv:DefInverse>` we saw that a  $2\times2$ matrix
+The word "determinant" already appeared in {numref}`Section %s <Sec:CrossProduct>` about the cross product. And in {numref}`Subsection %s <Subsec:MatrixInv:DefInverse>` we saw that a $2\times2$ matrix
 $A = \left[\begin{array}{cc} a & b \\ c & d\end{array}   \right]$ is invertible if and only if
 
 :::{math}
@@ -63,11 +63,13 @@ $$
 
 Thus we embed the plane into $\R^3$ as the $x_1$-$x_2$-plane. See {numref}`Figure %s <Fig:DetGeometric:OrientedArea1>`.
 
-::::{figure} Images/Fig-DetGeometric-OrientedArea1.svg
+```{applet}
+:url: det_geometric/orientedarea1
+:fig: Images/Fig-DetGeometric-OrientedArea1.svg
 :name: Fig:DetGeometric:OrientedArea1
 
 Oriented area
-::::
+```
 
 So we embed the plane $\R^2$ as the $x$-$y$-plane in $\R^3$.
 
@@ -141,7 +143,7 @@ $$
 
 where $\varphi$ is the angle from $\vect{u}$ to the left (= counterclockwise) to $\vect{v}$.
 
-We see that $ad-bc$ is equal to the area of the parallelogram if the directed angle from $\vect{u}$ to $\vect{v}$ is less then $\pi$, and minus this area if the angle is between $\pi$ and $2\pi$.  We call this the *signed area*.
+We see that $ad-bc$ is equal to the area of the parallelogram if the directed angle from $\vect{u}$ to $\vect{v}$ is less then $\pi$, and minus this area if the angle is between $\pi$ and $2\pi$. We call this the _signed area_.
 
 ::::::
 
@@ -289,28 +291,45 @@ $\det{(\vect{u},\vect{v}+\vect{w})} = \det{(\vect{u},\vect{v})} + \det{(\vect{u}
 </li>
 </ol>
 
-is the most interesting one. The two pictures of {numref}`Figure %s <Fig:DetGeometric:Linearity>` tell the story.
+is the most interesting one. The two pictures of {numref}`Figure %s <Fig:DetGeometric:Linearity1>` and {numref}`Figure %s <Fig:DetGeometric:Linearity2>` tell the story.
 
-::::{figure} Images/Fig-DetGeometric-SumRule.svg
-:name: Fig:DetGeometric:Linearity
+::::{figure}
+:name:
 
-The sum rule in a picture. Note, this is a **2D picture**.
 ::::
 
-In the picture on the left, both $(\vect{u},\vect{v})$ and $(\vect{u},\vect{w})$ are positively oriented.
+```{applet}
+:url: det_geometric/linearity_one
+:fig: Images/Fig-DetGeometric-SumRule.svg
+:name: Fig:DetGeometric:Linearity1
+:position: 2,2
+
+The sum rule in a picture with $(\vect{u},\vect{w})$ positively oriented. Note, this is a **2D picture**.
+```
+
+In the picture, both $(\vect{u},\vect{v})$ and $(\vect{u},\vect{w})$ are positively oriented.
 So there
 
 $$
 \begin{array}{lcl}
 \det{(\vect{u},\vect{v})} + \det{(\vect{u},\vect{w})} &=&
 \text{area}(OABC) + \text{area}(CBDE) \\
-&=& \text{area}(OADE) = \det{(\vect{u},\vect{v}+\vect{w})},
+&=& \text{area}(OADE) = \det{(\vect{u},\vect{v}+\vect{w})}.
 \end{array}
 $$
 
-since the two triangles $OCE$ and $ABD$ are congruent, so have equal areas.
+Since the two triangles $OCE$ and $ABD$ are congruent and they have equal areas.
+
+```{applet}
+:url: det_geometric/linearity_two
+:fig: Images/Fig-DetGeometric-SumRule.svg
+:name: Fig:DetGeometric:Linearity2
+:position: 2,2
+
+The sum rule in a picture with $(\vect{u},\vect{w})$ negatively oriented. Note, this is a **2D picture**.
+```
 
-In the picture on the right, the orientation of $(\vect{u},\vect{v})$ is positive, the orientation of $(\vect{u},\vect{w})$ is negative, and the orientation of $(\vect{u},\vect{v}+\vect{w})$ is positive again.
+In this picture, the orientation of $(\vect{u},\vect{v})$ is positive, the orientation of $(\vect{u},\vect{w})$ is negative, and the orientation of $(\vect{u},\vect{v}+\vect{w})$ is positive again.
 So there
 
 $$
@@ -318,28 +337,27 @@ $$
 \det{(\vect{u},\vect{v})} + \det{(\vect{u},\vect{w})} &=&
 \text{area}(OAFC) - \text{area}(OBDA) \\
 &=& \text{area}(OAFC) - \text{area}(CGEF) \\
-&=& \text{area}(OAEG) = \det{(\vect{u},\vect{v}+\vect{w})}
+&=& \text{area}(OAEG) = \det{(\vect{u},\vect{v}+\vect{w})}.
 \end{array}
 $$
 
-since now the areas of $OAFC$ and $CGEB$ add up to the area of $OABC$, owing to the
-congruence of the triangles $OGC$ and $AEF$.
-
 There are more pairwise orientations to consider, but the idea is hopefully clear to you.
 
 ::::::
 
-##  3x3 determinants: volume and orientation
+## 3x3 determinants: volume and orientation
 
 Suppose $\vect{a}, \vect{b}, \vect{c}$ are three vectors in $\R^3$. For the moment, suppose they are linearly independent. So
 $\vect{a}, \vect{b}$ are not multiples of each other, and $\vect{c}$ is not in the plane spanned by $\vect{a}, \vect{b}$.
-Then the three vectors can be interpreted as three edges of a parallelepiped. 
+Then the three vectors can be interpreted as three edges of a parallelepiped.
 
-::::{figure} Images/Fig-DetGeometric-Paraped.svg
+```{applet}
+:url: det_geometric/paraped
+:fig: Images/Fig-DetGeometric-Paraped.svg
 :name: Fig:DetGeometric:Paraped
 
 Volume equals base area times height
-::::
+```
 
 ::::::{prf:proposition}
 :label: Prop:DetGeometric:VolumeParped
@@ -353,11 +371,11 @@ $$
 
 ::::::
 
-
+::::
 
 ::::::{prf:proof}
 
-Just as the area of a parallelogram can be computed as 'base length times height', the volume of a parallelepiped can be computed as 'base area times height'.  See {numref}`Figure %s <Fig:DetGeometric:Paraped>`.
+Just as the area of a parallelogram can be computed as 'base length times height', the volume of a parallelepiped can be computed as 'base area times height'. See {numref}`Figure %s <Fig:DetGeometric:Paraped>`.
 As base region we can take the parallelogram spanned by $\vect{a}$ and $\vect{b}$, and then the base area becomes
 
 $$
@@ -371,22 +389,19 @@ $$
 h =  \norm{\vect{c}}\,\cos(\theta),
 $$
 
-which is positive if $\vect{c}$  lies on the same side of  plane $\mathcal{P}$ as $\vect{a} \times  \vect{b}$, and negative if these vectors lie on opposite sides.
+which is positive if $\vect{c}$ lies on the same side of plane $\mathcal{P}$ as $\vect{a} \times  \vect{b}$, and negative if these vectors lie on opposite sides.
 
-Using the alternative characterization of the inner product as in 
+Using the alternative characterization of the inner product as in
 {prf:ref}`Prop:InnerProduct:DotProdGeometric` we derive that
 
 $$
  V = |h|\norm{\vect{a} \times  \vect{b}} =   \norm{\vect{c}}\,|\cos(\theta)|\,\norm{\vect{a} \times  \vect{b}} = |\vect{c}\ip(\vect{a} \times  \vect{b})|.
 $$
 
-
 ::::::
 
-
-Note that the expression $\vect{c}\ip(\vect{a} \times  \vect{b})$ 
-gives the actual volume of the corresponding parallelepiped if the vectors $\vect{a}, \vect{b},\vect{c}$ are oriented as in the righthand rule (as in {numref}`Figure %s <Fig:CrossProduct:RightHandRule>`)  and minus this volume otherwise.
- 
+Note that the expression $\vect{c}\ip(\vect{a} \times  \vect{b})$
+gives the actual volume of the corresponding parallelepiped if the vectors $\vect{a}, \vect{b},\vect{c}$ are oriented as in the righthand rule (as in {numref}`Figure %s <Fig:CrossProduct:RightHandRule>`) and minus this volume otherwise.
 
 ::::::{prf:proposition}
 :label: Prop:DetGeometric:CyclicPerm
@@ -444,7 +459,6 @@ $$
 
 The next proposition expresses a 3×3 determinant in terms of three 2×2 determinants. This expression will be the key to define the determinant of a general n×n matrix in the next section.
 
-
 ::::::{prf:proposition}
 :label: Prop:DetGeometric:ColExpand
 
@@ -537,7 +551,6 @@ $\det{I} =  \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
 
 ::::::
 
-
 ## Grasple Exercises
 
 ::::::{grasple}
@@ -560,16 +573,15 @@ $\det{I} =  \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
 :url: https://embed.grasple.com/exercises/fc111f55-9f43-4730-a1f9-e3b5f03069bd?id=93133
 :label: grasple_exercise_5_1_3
 :dropdown:
-:description: Area of triangle in the plane. 
+:description: Area of triangle in the plane.
 
 ::::::
 
-
 ::::::{grasple}
 :url: https://embed.grasple.com/exercises/2d846d56-3729-468e-80d8-74ec6d348719?id=93134
 :label: grasple_exercise_5_1_4
 :dropdown:
-:description: Volume of parallelepiped  (with  vertex at (0,0,0)).
+:description: Volume of parallelepiped (with vertex at (0,0,0)).
 
 ::::::
 
@@ -577,12 +589,10 @@ $\det{I} =  \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
 :url: https://embed.grasple.com/exercises/1053b23-02ab-4ffb-bcda-70f808a9910a?id=74408
 :label: grasple_exercise_5_1_5
 :dropdown:
-:description: Volume of parallelepiped  (with  vertex at (0,0,0)).
+:description: Volume of parallelepiped (with vertex at (0,0,0)).
 
 ::::::
 
-
-
 ::::::{grasple}
 :url: https://embed.grasple.com/exercises/edc50bb7-425e-4a76-927e-f20504128f5f?id=65686
 :label: grasple_exercise_5_1_6
@@ -591,8 +601,6 @@ $\det{I} =  \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
 
 ::::::
 
-
-
 ::::::{grasple}
 :url: https://embed.grasple.com/exercises/47e1fd08-2d8a-4c0d-816a-37e314707191?id=87501
 :label: grasple_exercise_5_1_7
@@ -607,4 +615,4 @@ $\det{I} =  \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{ar
 :dropdown:
 :description: Using a determinant to check whether four points lie in the same plane.
 
-::::::
\ No newline at end of file
+::::::
diff --git a/Chapter5/DeterminantsViaCofactors.md b/Chapter5/DeterminantsViaCofactors.md
index 10bc57cb031854785566d5218778b65d6392c923..31570f52d1324637420e03b68fd922adcc87236f 100644
--- a/Chapter5/DeterminantsViaCofactors.md
+++ b/Chapter5/DeterminantsViaCofactors.md
@@ -428,7 +428,7 @@ We need to know a little more about determinants to establish this connection wi
 ## The Determinant of the Transpose of a Matrix
 
 
-The last property that may be expected to hold as a consequence of {prf:ref}`Thm:DetCofactors:RowOrColumnExpansion` where the rows and the columns play exchangeable roles is the following.
+The last property that may be expected to hold as a consequence of {prf:ref}`Thm:DetCofactors:RowOrColumnExpansion` where the rows and the columns play interchangeable roles is the following.
 
 ::::::{prf:proposition}
 :label: Prop:DetCofactors:DetTranspose
diff --git a/Chapter6/ComplexEigenvalues.md b/Chapter6/ComplexEigenvalues.md
index 3254aa401e6ef7917175886300ccb9fbc982e4bd..45761f7dd8cec3f86ccadae6c6c1de96db00e0a7 100644
--- a/Chapter6/ComplexEigenvalues.md
+++ b/Chapter6/ComplexEigenvalues.md
@@ -40,13 +40,13 @@ $$
 $$
 
 where the row operation we invoke is: add the second row $(1+i)$ times to the first row.
-The blue 0 is the result of the calculation of
+The blue {\color{blue}0} is the result of the evaluation of
 
 $$
 -2 + (1+i)(1-i).
 $$
 
-We can read off a solution (i.e., complex eigenvector) &nbsp;
+We can read off a solution (i.e., complex eigenvector): &nbsp; 
 $\vect{v} = \left[\begin{array}{c}  -1+i \\1 \end{array}\right]$.
 
 To check that we have indeed an eigenvector is also slightly more involved than in the real case.
@@ -95,12 +95,10 @@ $$
  = (1+i)\vect{v},
 $$
 
-so the two vectors are complex multiples of each other, and hence are not linearly independent.
+so the two vectors are *complex* multiples of each other, and hence are not linearly independent.
 
 For the other eigenvalue we can proceed in the same manner and find (for instance) the eigenvector
-
-<!-- prettier-ignore -->
-$\vect{v}_2 =\left[\begin{array}{c} -2 \\ 1-i \end{array}\right] $.
+&nbsp; $\vect{v}_2 =\left[\begin{array}{c} -2 \\ 1-i \end{array}\right] $.
 
 :::::
 
@@ -170,9 +168,6 @@ $\overline{\lambda} = \alpha - \beta i$ is an eigenvalue too.
 <li>
 
 If $\vect{v} = \vect{u}+i\vect{w}$, where $\vect{u}$ and $\vect{w}$ are real vectors, is an eigenvector
-
-<BR>
-
 for $\lambda$, then
 $\overline{\vect{v}} = \vect{u}-i\vect{w}$ is an eigenvector for $\overline{\lambda}$.
 
@@ -423,7 +418,7 @@ Let us illustrate matters with the following example.
 
 The matrix $A = \left[\begin{array}{cc} 1 & -2 \\ 1 & 3  \end{array}\right]$
 of {prf:ref}`Ex:ComplexEV:FirstExample`
-has the eigenvalue $\lambda = \alpha + \beta i = 2 + i$.
+has the eigenvalues $\lambda_{1,2} = \alpha \pm \beta i = 2 \pm i$.
 An eigenvector for $2-i$ is given by
 
 $$
@@ -432,7 +427,7 @@ $$
  + i\left[\begin{array}{c} 0 \\ -1  \end{array}\right].
 $$
 
-We check the identity
+Let us establish the identity
 
 $$
 P^{-1}AP = \left[\begin{array}{cc} \alpha & -\beta \\ \beta & \alpha \end{array}\right] = \left[\begin{array}{cc} 2 & -1 \\ 1 & 2 \end{array}\right].
diff --git a/Chapter6/Diagonalizability.md b/Chapter6/Diagonalizability.md
index ff25373167721800fda3eb06b5e392301b15299d..866f3636d422daa2dbeaa3d9f8e55237f4636a63 100644
--- a/Chapter6/Diagonalizability.md
+++ b/Chapter6/Diagonalizability.md
@@ -88,7 +88,7 @@ $$
 \det{(A - \lambda I)} = \det{(PBP^{-1} - \lambda I)} = \det{(B - \lambda I)}.
 $$
 
-The second equality is shown in the following chain of identities.
+The second equality is proved by the following chain of identities.
 
 $$
 \begin{array}{rcl}
@@ -320,7 +320,7 @@ $$
 ## Diagonalizability
 
 ::::::{prf:definition}
-:label: Dfn:Eigenvalues:Diagonalizability
+:label: Dfn:Diagonalizable:Diagonalizability
 
 A matrix is $A$ is called **diagonalizable** if it is similar to a diagonal matrix. That means that a diagonal matrix $D$ and an invertible matrix $P$ exist such that
 
@@ -554,7 +554,7 @@ The matrix $R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ of {prf:ref}`Ex:E
 ::::::{prf:remark}
 :label: Rem:Diagonalizable:RotationCtd
 
-Things would be different if we would allow complex eigenvalues and eigenvectors. We will devote a separate section to this. And then it will appear that the matrix $R$ is **complex diagonalizable**.
+Things would be different if we would allow complex eigenvalues and eigenvectors. We will devote a separate section ({numref}`Section:ComplexEV`) to this. And then it will appear that the matrix $R$ is **complex diagonalizable**.
 
 ::::::
 
@@ -627,8 +627,81 @@ First we show that a diagonalizable matrix satisfies the two conditions.
 
 If $A$ is diagonalizable, then there must be $n$ independent eigenvectors. The sum of the dimensions $m_k$ of the eigenspaces $E_{\lambda_i}$, i.e., the sum of the geometric multiplicities must therefore be equal to $n$. Since the algebraic multiplicities are at least as large as the geometric multiplicities, the sum of the algebraic multiplicities must be $\geq n$. Since this sum cannot be larger, it means that the sum is equal to $n$. Thus all algebraic multiplicities must in fact be equal to the corresponding geometric multiplicities. This settles properties i and ii.
 
-Conversely, conditions i. and ii. immediately imply that there must be $n$ linearly independent eigenvectors.
-Namely, since eigenvectors for different eigenvalues are automatically linearly independent, bases for the eigenspaces put together give exactly $n$ linearly independent eigenvectors.
+Conversely, conditions i. and ii. immediately imply that there must be $n$ linearly independent eigenvectors. 
+The basic idea is that, since eigenvectors for different eigenvalues are automatically linearly independent, the bases for the eigenspaces put together give exactly $n$ linearly independent eigenvectors.  
+
+::::::
+
+
+::::::{dropdown} (More detailed) Proof of&nbsp;{prf:ref}`Thm:Diagonalizable:ThirdCharacterization`
+
+Suppose that the $n \times n$ matrix $A$ has only real eigenvalues,  say  $\lambda_1,\ldots,\lambda_k$, and that for each eigenvalue $\lambda_i$ the geometric multiplicity $m_i$  is equal to the algebraic multiplicity, so
+
+$$
+  \text{g.m.}(\lambda_i)  = m_i= \text{a.m.}(\lambda_i) , \quad i = 1, \ldots, k.
+$$
+
+Since the sum of the algebraic multiplicities is equal to  $n$, the
+ sum of the geometric multiplicities must be equal to $n$ too,
+ 
+$$
+  m_1 + m_2 + \ldots + m_k = n.
+$$ 
+ 
+For each $i$ let $\{\vect{v}^{(i)}_1, \ldots, \vect{v}^{(i)}_{m_i}\}$  be a basis for the eigenspace  $E(\lambda_i)$. If we can show that the union of all these bases is a basis for $\R^n$, we have a basis of eigenvectors for matrix $A$, and by   {prf:ref}`Prop:Eigenvalues:DiagbleVersusEigenvectors` $A$ is diagonalizable.  To this end it is sufficient to show that the total set
+
+$$
+  \left\{\vect{v}^{(1)}_1, \ldots, \vect{v}^{(1)}_{m_1}, .\,.\,.\,.\,., \vect{v}^{(k)}_1, \ldots, \vect{v}^{(k)}_{m_k} \right\}
+$$   
+
+is linearly independent.  So, suppose that 
+
+:::::{math}
+:label:  Eq:Diagonalizable:SumAllEigenvalues
+
+   \underbrace{c^{(1)}_1\vect{v}^{(1)}_1+ \cdots + c^{(1)}_{m_1}\vect{v}^{(1)}_{m_1}} \,+ \,\cdot\,\cdot\,\cdot\,\cdot\,\cdot\,
++ \,  \underbrace{c^{(k)}_1\vect{v}^{(k)}_1+ \cdots + c^{(k)}_{m_k}\vect{v}^{(k)}_{m_k}}  = \vect{0}.
+
+:::::
+
+If we introduce
+
+$$
+   \vect{y}_i = c^{(i)}_1\vect{v}^{(i)}_1+ \cdots + c^{(i)}_{m_i}\vect{v}^{(i)}_{m_i}, \quad i = 1,\ldots,k,
+$$
+
+
+we have that  
+
+$$
+   \vect{y}_1 + \vect{y}_2 +  \ldots + \vect{y}_k = \vect{0}.
+$$
+
+Since each  vector  $\vect{y}_i$ lies in the eigenspace  $E(\lambda_i)$,  and by {prf:ref}`Prop:Eigenvalues:IndepEigenvectors`  eigenvectors for different eigenvalues are linearly independent, it follows that 
+
+$$
+  \vect{y}_i = \vect{0}, \quad i = 1, \ldots, k.
+$$
+
+So we have for each underbraced term  in Equation {eq}`Eq:Diagonalizable:SumAllEigenvalues`
+
+$$
+  c^{(i)}_1\vect{v}^{(i)}_1+ \cdots + c^{(i)}_{m_i}\vect{v}^{(i)}_{m_i} = \vect{0}.
+$$
+
+Since the vectors in a set $\{\vect{v}^{(i)}_1, \ldots, \vect{v}^{(i)}_{m_i}\}$ were supposed to form a basis for the eigenspace $E(\lambda_i)$, they must be linearly independent. Thus it follows for the coefficients in the last sum  that 
+
+$$
+    c^{(i)}_1 = 0, \,\ldots\,, \, c^{(i)}_{m_i}= 0.
+$$
+
+This shows that {eq}`Eq:Diagonalizable:SumAllEigenvalues`  can only hold if all coefficients  are zero, and consequently the set
+
+$$
+   \{\vect{v}^{(1)}_1, \,\ldots, \,\vect{v}^{(1)}_{m_1}, \,\,.\,.\,.\,.\,.\,\,, \,\vect{v}^{(k)}_1, \,\ldots, \, \vect{v}^{(k)}_{m_k}\} 
+$$
+
+being a linearly independent set of $n$ vectors, will indeed be a basis consisting of eigenvectors (for $\R^n$).
 
 ::::::
 
@@ -637,7 +710,7 @@ We saw that there is a weak connection between eigenvalues and (non-)invertibili
 {prf:ref}`Prop:EigenValues:SingularMatrix` states: a matrix is singular if and only if it has the eigenvalue $0$.
 
 
-In exercise 6.3.11 below you are invited to investigate the connection (or no-connection) between diagonalizability and invertibility.
+In exercise 6.3.12 below you are invited to investigate the connection (or no-connection) between diagonalizability and invertibility.
 
 %::::::{exercise}
 %:label: Exc:Diagonalizable:Invertibility
diff --git a/Chapter7/GramSchmidt.md b/Chapter7/GramSchmidt.md
index 4c6c58fa6aba50cb0bb7ceec48ca82a0498a9185..66547dd697d0dd779d84e7a36919e48adf7bbd84 100644
--- a/Chapter7/GramSchmidt.md
+++ b/Chapter7/GramSchmidt.md
@@ -178,7 +178,9 @@ $$
   \dfrac{\vect{a}_{j+1}\ip\vect{b}_1}{\vect{b}_1\ip\vect{b}_1}\vect{b}_1  + \ldots + \dfrac{\vect{a}_{j+1}\ip\vect{b}_j}{\vect{b}_j\ip\vect{b}_j}\vect{b}_j.
 $$
 
-This makes $\vect{b}_{j+1}$ an element of $W_{j}$.
+This makes $\text{proj}_{W_j}(\vect{a}_{j+1})$    an element of $W_{j}$  and 
+$\vect{b}_{j+1} = \vect{a}_{j+1} - \text{proj}_{W_j}(\vect{a}_{j+1})$  an element of
+$W_{j+1}$.
 
 $\vect{b}_{j+1} \neq \vect{0}$ since we assumed that the vectors $\vect{a}_1, \ldots, \vect{a}_m$ are independent, so $\vect{a}_{j+1}$ is not in $W_j$.
 
@@ -187,7 +189,7 @@ Since all $\vect{b}_i$ with $i \leq j$ lie in $W_j$, this makes $\vect{b}_{j+1}$
 Hence if the vectors $\vect{b}_1, \ldots, \vect{b}_j$ are orthogonal,
 so are the vectors $\vect{b}_1, \ldots, \vect{b}_j, \vect{b}_{j+1}$.
 
-(By using mathematical induction the proof can be made an absolutely rigorous.)
+(By using mathematical induction the proof can be made absolutely rigorous.)
 
 ::::
 
@@ -199,7 +201,7 @@ so are the vectors $\vect{b}_1, \ldots, \vect{b}_j, \vect{b}_{j+1}$.
 
 ::::
 
-The following example shows what happens if the Gram-Schmidt construction also works for
+The following example shows what happens if the Gram-Schmidt construction is applied to
 a subspace $W = \text{span}\{\vect{a}_1, \ldots, \vect{a}_m\}$ where the vectors $\vect{a}_i$ are not linearly independent.
 
 ::::{prf:example}
@@ -208,7 +210,7 @@ a subspace $W = \text{span}\{\vect{a}_1, \ldots, \vect{a}_m\}$ where the vectors
 Let $W$ be the defined as the span of the set $\left\{\begin{bmatrix} 1 \\ -1   \\ 2 \\ 3    \end{bmatrix}, \begin{bmatrix} 2 \\ -2   \\ 4 \\ 6    \end{bmatrix}, \begin{bmatrix} 1 \\ -1   \\ 3 \\  4    \end{bmatrix}, \begin{bmatrix} 1 \\ 3   \\ 1 \\  2    \end{bmatrix} \right\}$.
 
 Let us denote the vectors by $\vect{a}_1, \ldots, \vect{a}_4$. As in the proof of the Gram-Schmidt process
-we use the notation $W_j$ for the span of the vectors $ \vect{a}\_1, \ldots, \vect{a}\_j$.
+we use the notation $W_j$ for the span of the vectors $ \vect{a}_1, \ldots, \vect{a}_j$.
 
 Just following the protocol we find
 
@@ -292,7 +294,6 @@ $$
 we found, in {prf:ref}`Ex:GramSchmidt:Orthogonalize`, the orthogonal basis
 
 $$
-
    \left\{\begin{bmatrix} 1 \\ 1   \\-1  \\1   \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\0 \\-2  \end{bmatrix},   \begin{bmatrix} 2 \\ 0   \\3  \\1  \end{bmatrix}\right\}.
 $$
 
diff --git a/Chapter7/LeastSquares.md b/Chapter7/LeastSquares.md
index 6240658f0cdb098049ba5ad07aa10c544f7827c1..2eac65c111bf54a618cb2c9fabcaf6371879da00 100644
--- a/Chapter7/LeastSquares.md
+++ b/Chapter7/LeastSquares.md
@@ -93,7 +93,7 @@ $$
 What we can do if the linear system is inconsistent, thus if
 
 $$
-  \vect{b} \neq \Span{\vect{a}_1, ... , \vect{a}_n},
+  \vect{b} \notin \Span{\vect{a}_1, ... , \vect{a}_n},
 $$
 
 is to try to find the _best approximation_ of the vector $\vect{b}$ with a vector in
diff --git a/Chapter7/OrthoBase.md b/Chapter7/OrthoBase.md
index ad1734b5f9fa7a675c6214b74aed512ed48a0c80..234ff1568da4f822342f4cc8f21c5c3d193a7f0e 100644
--- a/Chapter7/OrthoBase.md
+++ b/Chapter7/OrthoBase.md
@@ -222,8 +222,11 @@ $$
 
 It is worthwhile to compare this result to the formula for the projection of one vector on another given in {prf:ref}`Prop:InnerProduct:UniqueProjection`. What {prf:ref}`Thm:OrthoBase:OrthoDecomp` states is essentially this: if $V$ has an orthogonal basis $\vect{v}_{1},...,\vect{v}_{k}$, then the projection of any vector $\vect{u}$ onto $V$ is the sum of the projections of $\vect{u}$ on the $\vect{v}_{i}$'s. This is illustrated in {numref}`Figure %s <Fig:OrthoBase:DecompAs2Proj>`
 
-```{figure} Images/Fig-OrthoBase-DecompAs2Proj.svg
+```{applet}
+:url: ortho/decompas2proj
+:fig: Images/Fig-OrthoBase-DecompAs2Proj.svg
 :name: Fig:OrthoBase:DecompAs2Proj
+:position: -5.49,7.46,20.08
 
 A vector and its orthogonal projection on the subspace $V$. Note that this projection is the sum of the projections of $\vect{u}$ on the orthogonal basis $\vect{v}_{1},\vect{v}_{2}$.
 ```
diff --git a/Chapter7/OrthoComp.md b/Chapter7/OrthoComp.md
index 3fac31733ab33f928f712799c1f26190ad4b4d1c..5d3b39d5d5daf7057dd9c15640ebdde98213f8c9 100644
--- a/Chapter7/OrthoComp.md
+++ b/Chapter7/OrthoComp.md
@@ -108,7 +108,9 @@ Both examples are illustrated in {numref}`Figure %s <Fig:OrthoComp:OrthoComp>`.
 
 :::
 
-```{figure} Images/Fig-OrthoComp-OrthoComp.svg
+```{applet}
+:url: ortho/orthocomp
+:fig: Images/Fig-OrthoComp-OrthoComp.svg
 :name: Fig:OrthoComp:OrthoComp
 
 The orthogonal complement of a 1-dimensional subspace of $\R^{2}$ (left) and of a 2-dimensional subspace of $\R^{3}$ (right).
@@ -275,7 +277,9 @@ Putting $\vect{u}_{V}=c_{1}\vect{v}_{1}+\cdots+c_{k}\vect{v}_{k}$ and $\vect{u}_
 
 :::
 
-```{figure} Images/Fig-OrthoComp-OrthoDecomp.svg
+```{applet}
+:url: ortho/orthodecomp
+:fig: Images/Fig-OrthoComp-OrthoDecomp.svg
 :name: Fig:OrthoBase:OrthoDecomp
 
 A subspace $V$, a vector $\vect{u}$ and the orthogonal decomposition of $\vect{u}$ with respect to $V$.
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deleted file mode 100644
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-/guMKEOWto4/cwAAAABJRU5ErkJggg=="/>
-</g>
-</g>
-</g>
-</g>
-</g>
-</g>
-<g clip-path="url(#clip_13)">
-</g>
-</g>
-</g>
-</svg>
diff --git a/Chapter8/SingularValueDecomp.md b/Chapter8/SingularValueDecomp.md
index 4034c4f8f4434689f4f4226dd7dbc8646d31d7a6..e00ff011ba85f1dd46f37b1af096cdae5c7fc576 100644
--- a/Chapter8/SingularValueDecomp.md
+++ b/Chapter8/SingularValueDecomp.md
@@ -216,7 +216,7 @@ Lastly iii.  follows from  ii.  by transposing the matrix, and making use of iv.
 ::::
 
 
-The main goal is to show that a singular value decomposition always exists, and, how to construct one. Both answers are answered in reversed order in the next subsection
+The main goal is to show that a singular value decomposition always exists, and, how to construct one. Both questions will be answered in reversed order in the next subsection
 
 (Subsec:SVD:Existence)= 
 
diff --git a/_ext/applet.py b/_ext/applet.py
index 61554449ed2562b659e9fab1b6091b75acf7a55b..b71d662d3f2c05acb7b3c340495963146ad4d435 100644
--- a/_ext/applet.py
+++ b/_ext/applet.py
@@ -4,70 +4,76 @@ from docutils import nodes
 from docutils.parsers.rst import directives
 from sphinx.directives.patches import Figure
 
-from utils import parse_options, generate_style, ReviewStatus
+from utils import parse_options, generate_style
 
 
-DEFAULT_BASE_URL = 'https://openla.ewi.tudelft.nl/applet/'
+DEFAULT_BASE_URL = "https://openla.ewi.tudelft.nl/applet/"
 
 
 class AppletDirective(Figure):
-	option_spec = Figure.option_spec.copy()
-	option_spec.update({
-		'url': directives.unchanged_required,
-		'fig': directives.unchanged_required,
-		'title': directives.unchanged,
-		'background': directives.unchanged,
-		'autoPlay': directives.unchanged,
-		'position': directives.unchanged,
-		'isPerspectiveCamera': directives.unchanged,
-		'enablePan': directives.unchanged,
-		'distance': directives.unchanged,
-		'zoom': directives.unchanged,
-		'height': directives.unchanged,
-		'width': directives.unchanged,
-		'status': directives.unchanged
-	})
-	required_arguments = 0
+    option_spec = Figure.option_spec.copy()
+    option_spec.update(
+        {
+            "url": directives.unchanged_required,
+            "fig": directives.unchanged_required,
+            "title": directives.unchanged,
+            "background": directives.unchanged,
+            "autoPlay": directives.unchanged,
+            "position": directives.unchanged,
+            "isPerspectiveCamera": directives.unchanged,
+            "enablePan": directives.unchanged,
+            "distance": directives.unchanged,
+            "zoom": directives.unchanged,
+            "height": directives.unchanged,
+            "width": directives.unchanged,
+            "status": directives.unchanged,
+        }
+    )
+    required_arguments = 0
 
+    def run(self):
+        url = self.options.get("url")
+        fig = self.options.get("fig")
+        assert url is not None
+        assert fig is not None
 
-	def run(self):
-		url = self.options.get('url')
-		fig = self.options.get('fig')
-		assert url is not None
-		assert fig is not None
+        self.arguments = [fig]
+        self.options["class"] = ["applet-print-figure"]
+        (figure_node,) = Figure.run(self)
 
-		self.arguments = [fig]
-		self.options['class'] = ['applet-print-figure']
-		(figure_node,) = Figure.run(self)
+        # Generate GET params and inline styling
+        params_dict = parse_options(self.options)
+        params_dict["iframe"] = (
+            "true"  # To let the applet know its being run in an iframe
+        )
+        params = "&".join(
+            [f"{key}={quote(value)}" for key, value in params_dict.items()]
+        )
+        style = generate_style(
+            self.options.get("width", None), self.options.get("height", None)
+        )
 
-		# Generate GET params and inline styling
-		params_dict = parse_options(self.options)
-		params_dict['iframe'] = 'true'  # To let the applet know its being run in an iframe
-		params = '&'.join([f'{key}={quote(value)}' for key, value in params_dict.items()])
-		status = ReviewStatus.parse(self.options.get('status', ''))
-		style = generate_style(self.options.get('width', None), self.options.get('height', None), status)
-
-		base_url = os.environ.get('BASE_URL', DEFAULT_BASE_URL)
-		full_url = f'{base_url}{url}{"?" if params else ""}{params}'
-		applet_html = f'''
-			<div class="applet" style="{style}">
+        base_url = os.environ.get("BASE_URL", DEFAULT_BASE_URL)
+        full_url = f'{base_url}{url}{"?" if params else ""}{params}'
+        applet_html = f"""
+			<div class="applet" style="{style}; ">
 				<noscript class="loading-lazy">
 					<iframe src="{full_url}" allow="fullscreen" loading="lazy" frameborder="0"></iframe>
 				</noscript>
 			</div>
-		'''
-		applet_node = nodes.raw(None, applet_html, format='html')
+		"""
+        applet_node = nodes.raw(None, applet_html, format="html")
 
-		# Add applet as the first child node of figure
-		figure_node.insert(0, applet_node)
+        # Add applet as the first child node of figure
+        figure_node.insert(0, applet_node)
 
-		return [figure_node]
+        return [figure_node]
 
 
 def setup(app):
-	app.add_directive('applet', AppletDirective)
-	return {
-		'version': '0.1',
-		'parallel_read_safe': True,
-        'parallel_write_safe': True,
-	}
+    app.add_directive("applet", AppletDirective)
+    return {
+        "version": "0.1",
+        "parallel_read_safe": True,
+        "parallel_write_safe": True,
+    }
diff --git a/_ext/utils.py b/_ext/utils.py
index 8b9bf40c492428d62d010872851f25ae4a7346d7..a4c3f3e0f0fd9b7d1bd2fa674c61a879eff5d567 100644
--- a/_ext/utils.py
+++ b/_ext/utils.py
@@ -2,23 +2,7 @@ import enum
 from typing import Optional
 
 
-class ReviewStatus(enum.Enum):
-	UNREVIEWED = enum.auto()
-	IN_REVIEW = enum.auto()
-	REVIEWED = enum.auto()
-
-	@staticmethod
-	def parse(s: str):
-		match s:
-			case 'in-review':
-				return ReviewStatus.IN_REVIEW
-			case 'reviewed' | 'approved':
-				return ReviewStatus.REVIEWED
-			case _:
-				return ReviewStatus.UNREVIEWED
-
-
-def generate_style(height: Optional[str], width: Optional[str], status: ReviewStatus):
+def generate_style(height: Optional[str], width: Optional[str]):
 	'''
 	Given a height and width, generates an inline style that can be used in HTML.
 	'''
@@ -31,11 +15,6 @@ def generate_style(height: Optional[str], width: Optional[str], status: ReviewSt
 	if width:
 		styles += f'width: {width};'
 
-	if status == ReviewStatus.UNREVIEWED:
-		styles += 'border: dotted red;'
-	elif status == ReviewStatus.IN_REVIEW:
-		styles += 'border: dotted yellow;'
-
 	return styles
 
 
diff --git a/_static/extra_css_config.css b/_static/extra_css_config.css
index 3dc1c859dff5d6c2806fe1370596f9c0b86c5027..4c2801746cf2c0e125b60687342e6573106464cc 100644
--- a/_static/extra_css_config.css
+++ b/_static/extra_css_config.css
@@ -1,10 +1,20 @@
+.applet {
+  height: 500px;
+}
+
 .applet * {
   width: 100%;
   height: 500px; /* TODO: subject for discussion */
 }
+
 .applet-print-figure {
   display: none;
 }
+
+.applet iframe {
+  border-radius: 10px;
+}
+
 @media print {
   .applet iframe {
     display: none;