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

For instance:

Take . The following transformation matrix is applied,

Actually, it's all fine and dandy when I do the matrix arithmetic ( ), 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'):

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