Visualize a 2d Linear (Matrix) Transformation!
The "image" of a point after the transformation is what happens when you multiply the column vector containing the coordinates of that point by the matrix on the left. Drag the large blue dots to change the image of (1,0), labeled X', and (0,1), labeled Y'. Notice how the images of (1,1), (2,1), and (-1,-2) change as you change X' and Y'. Notice that the first column of the matrix is the coordinates of X' and the second column is Y'. Try rotating, reflecting, and stretching the image and watch how the matrix changes.