Help with transformation matrices involving multiple transformations Transformations matrices have always been sort of a weak area for me just because it always feels like I'm doing the process wrong.
The question I'm trying to solve is this:
Suppose we stretch the picture horizontally by a factor of 2, rotate it 45 degrees clockwise, shrink it horizontally by a factor of 3 and then reflect it through the y-axis 
My attempt at the question:
The first step's matrix would be: \begin{bmatrix}2&0\\0&1\end{bmatrix}
The second step's matrix would be:
\begin{bmatrix}\cos(-\pi/4)&-\sin(-\pi/4)\\\sin(-\pi/4)&\cos(-\pi/4)\end{bmatrix}
The third step's matrix would be:
\begin{bmatrix}1/3&0\\0&1\end{bmatrix}
The last step's matrix would be:
\begin{bmatrix}-1&0\\0&1\end{bmatrix}
I multiplied all the transformation matrices together in the order given and got a final answer of approximately:
\begin{bmatrix}-0.4714&1.4142\\0.2357&0.7071\end{bmatrix}
I just want to make sure what I did was correct. If I made an error somewhere in the middle, please let me know. I cannot understand transformation matrices for the life of me :(
 A: Have you heard of 3blue1brown? That YouTube channel is the only reason I understood linear algebra and more.
Basically all you do is follow where the basis vectors go. Very smoothly and patiently, you will get an answer.
https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
https://youtu.be/XkY2DOUCWMU
But if you want to confirm your math homework answers very quickly you could try this Desmos feature I dug up.
https://www.desmos.com/matrix
I won't compute the actual thing here cause Latex is stressful on a phone. But if this down here is what you worked out, then you'll be fine(matrix multiplication is inside out like function composition):
$\begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{3} & 0\\ 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \begin{bmatrix} 2 & 0\\ 0 & 1 \end{bmatrix}$
Don't despair so easily. With enough practice, patience and intuition in something especially sweet like linear algebra you will get better at math.
Edit: I'm not sure how you did the multiplication cause I got:
$\begin{bmatrix} -\frac{\sqrt{2}}{3} & -\frac{\sqrt{2}}{6}\\ -\sqrt{2} & \frac{\sqrt{2}}{2} \end{bmatrix}$
