Let $\displaystyle T: V \to V$ be the linear transformation given by $\displaystyle T(\vec x)=c \vec x$ where $\displaystyle c\in \mathbb{C}$. Demonstrate that T is unitary if and only if $\displaystyle |c|=1$.

If $\displaystyle V$ is a unidimensional vector space, demonstrate that the only unitary linear transformations are the one given previously. In particular, if $\displaystyle V$ is a unidimensional vector space, there exist only 2 orthogonal transformations: $\displaystyle T(x)=x$ and $\displaystyle T(x)=-x$.

Attempt: Not much. I want to prove $\displaystyle \Rightarrow$ first.

If T is unitary, then $\displaystyle T^{-1}(\vec x)=T^{\dagger}(\vec x)$. I don't know how to go further, I must show that this implies that $\displaystyle |c|=1$.

Hmm... $\displaystyle T^{-1} (\vec x)=\frac{\vec x}{c}=x^{\dagger} c^{\dagger}$.

Stuck here.