I have trouble understanding the following exercise so I would really appreciate any help you could give me:
Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Let $H$ be defined as $H:=I-2\dfrac{\mathbf{n}\mathbf{n}^T}{\|\mathbf{n}\|^2}$, where $I$ is the identity matrix:
a) Prove, that $V = \{ u\in \mathbb{R}^n : k^Tu = 0\}$ is a vector subspace in $\mathbb{R}^n$ . What is its dimension?
b) Prove, that the transformation matrix $H$ mirrors over the subspace of $V$ in $\mathbb{R}^n$ (you have to prove, that $Hk=-k$ and $Hv = v$ for every $v\in V$).
c) Prove, by an example, that $H^2 = I$, so $H^{-1}=H$.