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 $H^{-1}=H$.
Let's talk about (a). You've got to prove that V is a sub space of $\mathbb{R}^n$, that's the idea: you can assume that the vector $k$ is $e_1=\left( \begin{array}{c} 1\\ 0\\ .\\ .\\ .\\ 0\\ \end{array} \right)$ $\space$ (why?!) and prove that $V=$Span$\{e_i, $ with $i$ from $2$ to $n$$\}$. As you can imagine its dimension is $n-1$. I suggest you to call it $k$ orthogonal or simply $k^{\perp}=V$
(b)&(c) looks a bit disconnected by (a), my interpretation it that you've got to show a matrix $H$, $n \times n$, such that $H^2=Id$ and $Hk=-k$. Consider $\begin{pmatrix}-1 & 0 \\ 0 & Id_{n-1}\end{pmatrix}$ and try to prove it, ask for any more hints.