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

For instance:

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

$\displaystyle X'=T \cdot X$

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

Actually, it's all fine and dandy when I do the matrix arithmetic ($\displaystyle X=T^{-1}X'$), 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'):

$\displaystyle x' = x+2y$

$\displaystyle y' = 2x+y$

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