Chapter 5

Shew that the nofm defined by an mner product satisfies the paratlelogranu taw $$ \|u+v\|^{2}+\|u-v\|^{2}=2\|u\|^{2}+2\|v\|^{2} $$

Let \((V, \cdot)\) be a real Euclidean inner product space and denote the length of a vector \(=\sqrt{x+x}\). Show that two vectors \(u\) and \(v\) are orthogonal iff \(|u+v|^{2}=|u|^{2}+|v|^{2} .\)

$$ \mathrm{G}=\left[3_{2}\right]=\left[\begin{array}{ll} u_{r} & u_{\prime} \end{array}\right]=\left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & -1 & 1 \end{array}\right) $$ be the components of a rcal inner product with respect to a basis \(u_{1}, t_{2}, u_{3}\). Use Gram-Schmidt orthogonalzzation to find an orthonoi nal basis \(e_{1}, e_{2}, e_{3}\), expressed un terms of the vectors \(u_{1}\), and find the mdex of this uner product,