This applet helps us understand how multiplication by a matrix (or how a linear transformation) affects the direction. Drag the green arrow to change its direction and notice that the blue arrow will change as a result. The blue arrow is the result of Mu, where u is the green arrow. Notice that there are two places where the direction of the green arrow and the direction of the blue arrow coincide. These are [b]invariant[/b] directions: directions that do not change by matrix multiplication. The default matrix in this applet is {{1,2},{2,1}}, that is, the first row is 2,1 and the second row is 1,2. Try some other matrices. Is this behavior consistent with all matrices? To change the matrix, enter M={{a,b},{c,d}} in the input box at the bottom (for your choice of a,b,c,d.) Some matrices you might want to try: M={{1,2},{-2,1}} M={{1,2},{2,4}}