Solution to Exercise 3.22.
Mathematics is well and good, but nature keeps dragging us around by the nose.
—Albert Einstein (1879–1955)

Exercise 3.22.  Find the matrix of an affine transformation that maps P(0, 0, 1) to P'(0, 2, 1), Q(1, 0, 1) to Q'(2, 1, 1), and R(2, 3, 1) to R'(7, 9, 1).
Let  be the matrix of the affine transformation.

 implies . Hence, a₁₃ = 0 and a₂₃ = 2.

 implies . Hence, a₁₁ = 2 and a₂₁ = –1.

 implies . Hence, a₁₂ = 1 and a₂₂ = 3.

Therefore,