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 invariant 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}}