interpreting x versus y-axes in rotation matrix consider a matrix $M= \begin{bmatrix}x_{1}&x_{2}\\y_{1}&y_{2}\end{bmatrix}$. we can think of each column as a vector and visualise the matrix as a set of vectors drawn from the origin $(0,0)$: M
to the matrix $M$ we can apply the transformation matrix $T$ through matrix multiplication $MT$. for $M= \begin{bmatrix}-1&3\\2&2\end{bmatrix}$, we can apply the reflection matrix $T =  \begin{bmatrix}-1&0\\0&1\end{bmatrix}$, to get $MT = \begin{bmatrix}1&3\\-2&2\end{bmatrix}$ (reflected matrix's vectors in red): MxT
since we interpreted each column of $M$ to be $[x, y]$, i expected $T$ to reflect $M$'s vectors about the x-axis, not the y-axis. if the coordinate system is such that each column of a matrix is $[x, y]$, why is the vector's y-coordinate "reflected"?
 A: First, note that $T$ sends a vector $\pmatrix{x\\y}$ to $\pmatrix{-x\\y}$ which is conventionally called "a reflection across the $y$-axis."
Also, function composition is written so that "$MT$" means "Do $T$ first, then apply $M$ (as in $MT(v)) = M(T(v))$) as is the case with "normal" function composition $(f \circ g)(x) = f(g(x))$. That's why it's the composition $TM$ that reflects "$M$'s vectors" across the $y$-axis; $M$ gets a turn first, being closer to the input.
The composition $MT$, "do T first, then $M$," can be examined geometrically as well. Let me use the notation $e_1 = \pmatrix{1\\0}$ and $e_2 = \pmatrix{0\\1}$. Now, by "$M$'s vectors" I think you mean $Me_1$ and $Me_2$. Do you notice how $MTe_1$ is the negative of $Me_1$? That's because $T$ sends $e_1$ to its negative; $Te_1 = -e_1$. Since the transformations are linear, we have then that
$$  MTe_1 = M(Te_1) = M(-e_1) = -Me_1. $$
Also, because $T$ fixes $e_2$ (that is, $Te_2 = e_2$), we have that $MTe_2 = M(Te_2) = M(e_2) = Me_2$.
If you want $MT$ to mean "do $M$ first" you have to let matrices "act" on the right; you have to write your vector $v$ as a row vector to the left of your matrices (so that the matrix is on the right; then, the one acting first is the one furthest left, closest to the input vector).
