diff --git a/Chapter1/Inner_Product.md b/Chapter1/Inner_Product.md index dc5c86f69d320cbc8d941aba595fa22e289e11ef..7fe773666d6882c20081bdf035ddcef6551e01e1 100644 --- a/Chapter1/Inner_Product.md +++ b/Chapter1/Inner_Product.md @@ -403,7 +403,7 @@ Again it's a chain of identities using basic properties of the dot product. $$ \begin{array}{rcl} \norm{\mathbf{v}_1+\mathbf{v}_2}^2 + \norm{\mathbf{v}_1-\mathbf{v}_2}^2&=& (\mathbf{v}_1+\mathbf{v}_2)\cdot(\mathbf{v}_1+\mathbf{v}_2) + (\mathbf{v}_1-\mathbf{v}_2)\cdot(\mathbf{v}_1-\mathbf{v}_2) \\ - &=& \mathbf{v}_1\cdot\mathbf{v}_1 +2\mathbf{v}_1\cdot\mathbf{v}_2 + \mathbf{v}_1\cdot\mathbf{v}_2 + \mathbf{v}_1\cdot\mathbf{v}_1 -2\mathbf{v}_1\cdot\mathbf{v}_2 + \mathbf{v}_2\cdot\mathbf{v}_2 \\ + &=& \mathbf{v}_1\cdot\mathbf{v}_1 +2\mathbf{v}_1\cdot\mathbf{v}_2 + \mathbf{v}_2\cdot\mathbf{v}_2 + \mathbf{v}_1\cdot\mathbf{v}_1 -2\mathbf{v}_1\cdot\mathbf{v}_2 + \mathbf{v}_2\cdot\mathbf{v}_2 \\ &=& 2\,\mathbf{v}_1\cdot\mathbf{v}_1 +2\,\mathbf{v}_2\cdot\mathbf{v}_2 \\ &=& 2 (\norm{\mathbf{v}_1}^2 + \norm{\mathbf{v}_2}^2). \end{array}