I don't really get 'some' transformation matrices. The ones that have a number other than 0 in a_{12} or a_{21}.

For instance:

Take y=x^2. The following transformation matrix is applied,

X'=T \cdot X

\iff \begin{bmatrix} x' \\ y' \end{bmatrix}=\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

Actually, it's all fine and dandy when I do the matrix arithmetic ( X=T^{-1}X'), but I find it kinda hard to see what's going on. When the transformation is turned into equations, with (x,y) mapping onto (x',y'):

x' = x+2y

y' = 2x+y

Then how would I find the new equation without matrices? Thanks!