# Matrix rotation, projection, and reflection

What 3 by 3 matrices represent the transformations that

a) project every vector onto the $x-y$ plane?

b) reflect every vector through the $x-y$ plane?

c) rotate the $x-y$ plane through 90 degrees, leaving the z-axis alone?

d) rotate the $x-y$ plane, then $x-z$, then $y-z$, through 90 degrees?

I am very confused as to how to approach these problems. When dealing with just 2x2 matrices, I know that the rotation matrix is just $\begin{bmatrix} cos\theta & -sin\theta\\ sin\theta&cos\theta \end{bmatrix}$, and if I wanted to rotate something onto the $x$-axis, I would let $\theta =0$ and the transformational matrix would just be $\begin{bmatrix} 1 & 0\\ 0&1 \end{bmatrix}$. The 2 by 2 projection and reflection matrices $\begin{bmatrix} c^{2} & cs\\ cs&s^{2} \end{bmatrix}$, $\begin{bmatrix} 2c^{2}-1 & 2cs\\ 2cs&2s^{2}-1 \end{bmatrix}$, respectively. But when the 3rd dimension is introduced, I don't know how to approach these problems anymore. Could anyone walk me through this?

• Not an answer, but a hint is to read up on Householder matrices. – Shahab Mar 13 '14 at 7:34

## 1 Answer

If $e_1, e_2, \ldots, e_n$ is the standard basis of $\mathbb R^n$ and $f\colon \mathbb R^n \to \mathbb R^m$ is a linear transformation then the matrix that represents $f$ is the matrix whose columns are $f(e_1), f(e_2), \ldots, f(e_n)$.