Finding a matrix $A$ for each of the given conditions Represent each of the following three functions $f : {\bf R}^2 → {\bf R}^2$ as a matrix-vector product $f(x) = Ax$.
(a) $f(x)$ is obtained by reflecting $x$ about the $x_1$ axis.
(b) $f(x)$ is $x$ reflected about the $x_1$ axis, followed by a counterclockwise rotation of 30 degrees.
(c) $f(x)$ is $x$ rotated counterclockwise over 30 degrees, followed by a reflection about the $x_1$ axis.
 A: Take the standard base of $\mathbb{R}^2$, $(b_1, b_2) = (\binom{1}{0}, \binom{0}{1})$. Then calculate $f(b_1)$ and $f(b_2)$. $f(b_1)$ will be the left column and $f(b_2)$ will be the right column of your Matrix $A$.
A: It suffices to check what happens to the standard basis vectors. For (a), note that $f(1,0) = (1,0)$, which tells us that the first column of $A$ will be $(1,0)$.  Similarly, $f(0,1) = (0,-1)$, which tells us that the second column of $A$ will be $(0,-1)$.  All together, we find that
$$
A = \pmatrix{1&0\\0&-1}
$$
Note also that if $B$ is the matrix describing the transformation in (b), then the matrix product $AB$ will be the answer to (c).
A: Let $x = [x_1, x_2]^T$



*

*We want $x$ to end up $[x_1, -x_2]^T$


through $$T(x) = Ax$$ for some $A$.
Hence solve
$$\begin{bmatrix}
a & b\\ 
c & d
\end{bmatrix}[x_1, x_2]^T=[x_1, -x_2]^T$$
to get
$$\begin{bmatrix}
1 & 0\\ 
0 & -1
\end{bmatrix}$$

2,3
Firstly, to get a counterclockwise rotation only, refer to Larson Edwards Falvo - Elementary Linear Algebra Section 6.1



Thus,
$$\begin{bmatrix}
\cos(30^o) & -\sin(30^o)\\ 
\sin(30^o) & \cos(30^o)
\end{bmatrix}[x_1, x_2]^T = [\sqrt{x_1^2+x_2^2}\cos(30^{o} + \alpha), \sqrt{x_1^2+x_2^2}\sin(30^{o} + \alpha)]^T$$
where $$\cos(\alpha) = \frac{x_1}{\sqrt{x_1^2+x_2^2}}$$
Exercise: Which is for 2? For 3?
$$T(x) = \begin{bmatrix}
1 & 0\\ 
0 & -1
\end{bmatrix}\begin{bmatrix}
\cos(30^o) & -\sin(30^o)\\ 
\sin(30^o) & \cos(30^o)
\end{bmatrix}[x_1, x_2]^T$$
$$T(x) = \begin{bmatrix}
\cos(30^o) & -\sin(30^o)\\ 
\sin(30^o) & \cos(30^o)
\end{bmatrix}\begin{bmatrix}
1 & 0\\ 
0 & -1
\end{bmatrix}[x_1, x_2]^T$$
