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 2×2 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?
Best 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 $f(e_1), f(e_2), \ldots, f(e_n)$.