I'm reviewing for my final and I encountered a statement that puzzles me. Let $u,v$ be vectors in $\Bbb R^n$ that are expressed in column form, and let $A$ be an invertible $n\times n$ matrix. Then we can express the Euclidean inner product on $\Bbb R^n$ to be $\langle u,v\rangle = Au \cdot Av$.
Why does $A$ have to be an invertible matrix? I had a review question asking if everything but the invertibility property of $A$ held, if the statement was true, and the solutions in the back says that it was false.
Thanks in advance!
Best Answer
By definition an inner product must have the property that $\langle u,u\rangle=0$ if and only if $u=0$.
Suppose $\langle\cdot,\cdot\rangle$ is an inner product, $A$ is not invertible, and $\langle u,v\rangle_A:=\langle Au,Av\rangle$ is a bilinear form; can you show that there is a nonzero vector $u$ for which $\langle u,u\rangle_A=0$? $\color{White}{\mathrm{Hint}:Au=0\implies \langle u,u\rangle_A=0.}$