Are orthogonal operators always isomorphisms?

Question

I need to show the following:

Let $V$ be a finite dimensional vector space with inner product, if $T$ is orthogonal, show that T is injective and surjective

I think it is injective because T preserves inner product but i am not so sure if it is the right wya to prove it

Thanks for your help!

Answer 1

Assume $u\in V$ is such that $Tu=0$. The operator being orthogonal one has

$$(u,u)=(Tu,Tu)=0$$

This means $u=0$ and $T$ is injective and we’re done for bijectivity because $T$ is a linear operator in a finite dimensional vector space.

Answer 2

For $u,v\in V$ so that $T(u)=T(v)$, we have that $T(u-v)=0,$ so that $\langle u-v,u-v\rangle=\langle T(u-v),T(u-v)\rangle=0$. This means that $u-v=0$, thus $u=v$, so that $T$ is injective. On the other hand, the image of an injective linear transformation is a subspace of the codomain $V$ with dimension equal to the dimension of the domain, but since the dimension of the domain is just the dimension of the whole of $V$, $T$ is surjective.