For any two vectors $v_1$ and $v_2$, $\langle v_1, v_2 \rangle = \langle T (v_1), T (v_2) \rangle$ if $T$ is norm-preserving.

Question

A linear transformation $T : V \to V$ is said to be norm-preserving for a given inner product if for all vectors $v$ in $V$ the following is true: $\langle v,v \rangle = \langle T(v),T(v) \rangle$.

Prove that if $T$ is a norm-preserving transformation then for any vectors, $v_1$ and $v_2$ in ANY inner product space $V$, $\langle v_1, v_2 \rangle = \langle T (v_1), T (v_2) \rangle$.

I don’t get what I’m supposed to do, but I do know that the transformation is norm-preserving on the vector $v_1 + v_2$.

Use Polarization identity: https://en.wikipedia.org/wiki/Polarization_identity — Kavi Rama Murthy, May 29 '20 at 08:04
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, May 29 '20 at 08:08
Another related post: $T$ preserves inner product iff $|T\alpha |=|\alpha|$ — Martin Sleziak, Dec 03 '21 at 10:46

score 1 · Answer 1 · answered May 29 '20 at 08:08

1

Notice that for any $v, w \in V$, $$\langle v + w, v + w \rangle = \langle v , v \rangle + 2 \langle v, w \rangle + \langle w, w \rangle$$

So that:

$$\langle v, w \rangle = \frac{\langle v + w, v + w \rangle - \langle v , v \rangle - \langle w , w \rangle}{2}$$

If $T$ is norm preserving, it preserves all the terms on the right hand side, and since it is linear we have:

$$\frac{\langle T(v + w), T(v + w) \rangle - \langle T(v) , T(v) \rangle - \langle T(w) , T(w) \rangle}{2} = \langle T(v), T(w) \rangle = \langle v, w \rangle $$

answered May 29 '20 at 08:08

Matheus Andrade

6,773

1

This is of course assuming the space is over the real numbers. Over the complex numbers this does not work. – J. De Ro May 29 '20 at 08:19
Good point! You're most certainly correct. But the complex case is just a little bit more complicated. – Matheus Andrade May 29 '20 at 08:21
As pointed out above, the polarization identity can then be used instead. – J. De Ro May 29 '20 at 08:22

For any two vectors $v_1$ and $v_2$, $\langle v_1, v_2 \rangle = \langle T (v_1), T (v_2) \rangle$ if $T$ is norm-preserving.

1 Answers1