How to find the standard matrix A for T 
Let $T: \mathbb R^2 \rightarrow \mathbb R^2$ be the linear transformation that first rotates points clockwise through $30$ degrees and then reflects points through the line $y = x$
Find the standard matrix $A$ for $T$

I know that the standard matrix will be a $2x2$ matrix obviously.
I'm not sure how exactly to get the entries though to fill its columns
 A: In general the matrix of a rotation is a matrix of the form:
$R_1 = \begin{pmatrix}
  cos(\theta) & -sin(\theta) \\
  sin(\theta) & cos(\theta) 
 \end{pmatrix}$
while the matrix of a reflection is of the form:
$R_2 = \begin{pmatrix}
  cos(2\phi) & sin(2\phi) \\
  sin(2\phi) & -cos(2\phi) 
 \end{pmatrix}$
Where $\theta$ is the angle of the rotation so $\frac{\pi}{6}$, $\phi$ the angle of the line of the reflection, hence $\phi = \frac{\pi}{4}$
So you will have
$R_1 = \begin{pmatrix}
  \frac{\sqrt{3}}{2} & -\frac{1}{2} \\
  \frac{1}{2} & \frac{\sqrt{3}}{2}
 \end{pmatrix}$ for the rotation, and also $R_2 = \begin{pmatrix}
  0 & 1 \\
  1& 0 
 \end{pmatrix}$ for the reflection. Hence the matrix $A$ is $A = R_2R_1$.
As suggested, here a little explaination. You know that the any linear transformation is uniquely determined by what it actually does to the standard basis vectors. In this case they are $e_1 = (1,0)$ and $e_2 = (0,1)$. These are unit vectors on the Cartesian Plane, lying on the $x$ and $y$ axis. Imagine that you rotate them, so that the angle between them is still $\frac{\pi}{2}$. Image you rotate them of an angle $\theta$. Then clearly the linear transformation would give you $T(e_1) = (\cos(\theta),\sin(\theta))^T$ by trigonometry.
Applying the same concept to $e_2$ and then same method but different results for the reflection, you'll get the desidered matrices.
A: Memorizing the form of the rotation matrix and the reflection matrix probably aren't the best way to do this at your level.  That comes after you've done a bunch of these problems.  What you want to do is figure out where the standard basis vectors get mapped to.  I.e. you want to figure out -- geometrically if possible -- the vectors $T(1,0)$ and $T(0,1)$.  Then, writing those as column vectors, your matrix will be $$A = \pmatrix{T(1,0) & T(0,1)}$$
For some exposition on why the matrix has that form see this answer.
Let me know if you need help figuring out how to get $T(1,0)$ and $T(0,1)$, but try it yourself first.
