Linear Algebra Done Right · Problem 6A.5

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Suppose

V

is a real inner product space. ((a)) Show that

\langle u + v, u - v \rangle = \|u\|^2 - \|v\|^2

for every

u, v \in V

. ((b)) Show that if

u, v \in V

have the same norm, then

u + v

is orthogonal to

u - v

. ((c)) Use (b) to show that the diagonals of a rhombus are perpendicular to each other.