How do you find the matrix corresponding to a transformation?

Let's say that T is a transformation of the two dimensional plane. Remember that we have the two standard unit vectors (1,0) and (0,1). These are, respectively, the unit vectors pointing in the positive direction on the x-axis and the y-axis. We first look at what the transformation does to these two vectors. This gives us two new vectors T(1,0) and T(0,1) which form the columns of the matrix corresponding to the transformation T.

For example, if T is the reflection in the y-axis we get the following. Since we reflect in the y-axis, all points on the y-axis stay fixed and so T(0,1) = (0,1). On the other hand, by reflection (1,0) in the y-axis we get the point (-1,0). Therfore, the matrix has columns (-1,0) and (0,1).