Chapter 6 - Orthogonality and Least Squares - 6.1 Exercises - Page 339: 24

See solution

$||\vec{u}+\vec{v}||^2=(\vec{u}+\vec{v})\cdot(\vec{u}+\vec{v})=\vec{u}^T\vec{u}+2\vec{u}^T\vec{v}+\vec{v}^T\vec{v}$ $||\vec{u}-\vec{v}||^2=(\vec{u}-\vec{v})\cdot(\vec{u}-\vec{v})=\vec{u}^T\vec{u}-2\vec{u}^T\vec{v}+(-\vec{v})^T(-\vec{v})=\vec{u}^T\vec{u}-2\vec{u}^T\vec{v}+\vec{v}^T\vec{v}$ $||\vec{u}+\vec{v}||^2+||\vec{u}-\vec{v}||^2=\vec{u}^T\vec{u}+2\vec{u}^T\vec{v}+\vec{v}^T\vec{v}+\vec{u}^T\vec{u}-2\vec{u}^T\vec{v}+\vec{v}^T\vec{v}=2\vec{u}^T\vec{u}+2\vec{v}^T\vec{v}=2\vec{u}\cdot\vec{u}+2\vec{v}\cdot\vec{v}=2||\vec{u}||^2+2||\vec{v}||^2$