Manipulating inner products

Question

I have a function $f:V \rightarrow V$, $f(v) = v − 2 \langle u, v\rangle u$, where $u$ is a fixed unit vector.

I want to show that this $\langle f(v), f(w) \rangle= \langle v,w \rangle$ but I am unsure on how to manipulate the inner products. Can someone show this step by step?

Answer 1

We know that

$$f(v) = v − 2 \langle u, v\rangle u$$

So we plug that in:

$$\langle f(v), f(w) \rangle = \langle v − 2 \langle u, v\rangle u \rangle, w − 2 \langle u, w\rangle u \rangle $$

And now from the linearity of the dot product:

$$\langle v − 2 \langle u, v\rangle u \rangle, w − 2 \langle u, w\rangle u \rangle =\\ \langle v, w \rangle + \langle - 2\langle u, v \rangle v, w \rangle + \langle v, -2 \langle u, w \rangle w \rangle + \langle -2\langle u, v \rangle v, -2 \langle u, w \rangle w \rangle $$

Now can you take the constants out of the dot products to show that everything but the first dot product cancels?

Answer 2

Hint:

from the definition show that $\langle f(v),f(v) \rangle=\langle v,v\rangle$, so $f$ is an isometry and conserve angles.