# Subspaces, transformation matrices exercise

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 $H^{-1}=H$.

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## 1 Answer

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.

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I thought we can show that something is a subspace by proving it's closed under addition and scalar multiplication? Ok, I understand that e1 can be k, because it is a standard vector. Also, what does the notation Id<sub>n-1</sub>mean? Identity matrix of n-1? How do you multiply that? – Trom Nov 24 '12 at 20:02
1) Ye, that's right, I was just following an other way (and showing the moral behind the exercise). If you want to follow the way of direct verify you won't find any difficulty and it's exactly what you said. 2) It's just a notation. It's a square $n$ matrix which has an $Id_{n-1}$ inside. – Ivan Nov 25 '12 at 7:20