Apply the same matrix to three arrowsThe final state of the matrix transformation with rows (2,1) and (1,2), drawn by hand. The grid has been stretched along the upward diagonal. The teal vector x=(3,1) changes to Mx=(7,5), changing its direction. The blue vector q1=(1,-1) is unchanged. The orange vector q2=(1,1) becomes (3,3), three times its original length on the same line. Dashed arrows mark the starting positions of x and q2. The ordinary coordinate axes stay fixed, with equal scales. The animation moves from each input to its output under one application of M. Almost everybody moves. Two directions stay on their lines. The teal arrow turns. Blue and orange stay on their dashed lines. x1​ x2​ -2 2 4 6 8 10 2 4 6 0 Mx Mq2​ = 3q2​ Mq1​ = q1​ Output Mx: (7, 5) q1​: unchanged q2​: length × 3 Dashed arrows: where x and q2​ started. The axes stay put.
100%
BeforeAfter M

Follow the arrows from where they start to where M sends them. This is one application of M.

The same M transforms the whole grid. The ordinary axes stay fixed so we can compare before and after. Dashed arrows mark where x and q₂ began; q₁ doesn't move at all.

Figure notes

  1. The wobble in the hand-drawn grid is drawn on the plane itself, so M transforms the wobbles along with everything else. That's what a linear map does to any drawing you make on the plane. (The arrow tips sit exactly where the math says, though.) ↩