I have done the work however I wanted to make sure my answers were correct.
If you prefer the picture of the question: Picture
Problem (still getting the hang of mathjax I tried to make it as neat as possible):
Consider the linear transformation $T:\mathbb R^2 \to \mathbb R^2$ that first rotates a vector with $\pi/4$ radians clockwise and then projects onto the $x_2$ axis
(a) Find $$ T\begin{pmatrix} 1 \\ 1 \\ \end{pmatrix} $$
(b) Find the standard matrix for $T$.
(c) Is $T$ onto (surjection)?
For (a) with some error and help I got to the answer of
$$ \left[ \begin{array}{cc|c} 0\\ 0 \end{array} \right] $$
I am confused how I should be getting (b). Is my answer correct?
$$ \left[ \begin{array}{cc|c} \cos(\pi/4)&-\sin(\pi/4)\\ \sin(\pi/4)&\cos(\pi/4) \end{array} \right] $$ times
$$ \left[ \begin{array}{cc|c} 0&0\\ 0&1 \end{array} \right] $$
After I multiplied those two together I got
$$ \left[ \begin{array}{cc|c} 0&\sqrt(2)/2\\ 0&\sqrt(2)/2 \end{array} \right] $$
Would this be the standard matrix for $T$?
Also can anyone lead me in the direction of how I can determine if this is a surjection or not?
Thank you!
You made two mistakes:
The matrix you wrote, i.e.,
$$ \left[ \begin{array}{cc|c} \cos(\pi/4)&-\sin(\pi/4)\\ \sin(\pi/4)&\cos(\pi/4) \end{array} \right] $$
is the matrix for rotation by $\frac\pi4$ anti-clockwise, so it is not the right matrix.
Second, you then multiplied that matrix $R$ by the projection matrix $P$, but that means that the matrix you ended up with, $R\cdot P$, acts on the vector $x$ like so: $$(R\cdot P)\cdot x = R\cdot(P\cdot x)$$ which means you first project the vector, then you rotate it. This is incorrect.