$$\begin{pmatrix}1 & 3\\\ 5 & 6\end{pmatrix} * \begin{pmatrix}x \\ y \end{pmatrix} $$
I have just recently started to learn matrices so I apologize if I make any mistakes explaining my question.
A 2x2 matrix like the one above I learnt represents the linear transform of some vectors (the second matrix above x and y) when its multiplied with the second matrix. I sort of understand how this works.
Since a 2d vector can be represented as x $\widehat{i}$ + y $\widehat{j}$, basically saying 'go x times to the right and y times up', with the transformed matrix, im saying 'go 4x times to the right and 11y times up, that will be your transformed vector'
But in the case of something like, $$\begin{pmatrix}1 & 3\\\ 5 & 6\end{pmatrix} * \begin{pmatrix}x & x1 \\ y & y1 \end{pmatrix} $$
x and y is a vector right? and x1 and y1 is a second set of vectors right? So here im using $$\begin{pmatrix}1 & 3\\ 5 & 6\end{pmatrix} to transform these vectors and return me two transformed vectors rather than just one previously
But that leaves me confused. Does that mean even $$\begin{pmatrix}1 & 3\\\ 5 & 6\end{pmatrix}$$ is just 2 vectors? And if those 2 vectors are considered a transform, does that mean even $$\begin{pmatrix}x \\ y \end{pmatrix} $$ is a transform too? Any and all matrices are transforms, thus if any and all matrices represent a vector, all vectors are transforms by that logic? Which brings to my question...
...So a matrix-matrix multiplication is just a transform multiplied by a transform? Im not able to wrap my head around that idea that every matrix multiplication with another matrix is just 2 transforms multiplied by each other.
I apologize if my formatting for the matrices was not correct, I do not know how to get them to have straight brackets as im new to this kind of formatting