- #1

Mr Davis 97

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## Homework Statement

T/F: If ##T: \mathbb{R}^n \rightarrow \mathbb{R}^m## is a linear transformation and ##n>m##, then the function ##\langle v , w \rangle = T(v) \cdot T(w)## is an inner product on ##\mathbb{R}^n##

## Homework Equations

## The Attempt at a Solution

The first three axioms of the inner product are straightforward. However, I am not sure how to show that ##\langle v , v \rangle = 0## iff ##v0##.

Actually, maybe I have something. If ##\langle v , v \rangle = 0## then ##T(v) \cdot T(v) = ||T(v)||^2 \implies ||T(v)|| = 0 \implies T(v) = 0##. Thus, v is zero iff the null space of T is only zero. This is only the case when T is invertible. However, n > m, so T can't be invertible. Thus, v doesn't have to be 0, and thus we don't have an inner product space. Is this on the right track?

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