Solution to Exercise 3.36.
The only way to learn mathematics is to do mathematics..
—Paul Halmos (1916-2006)

Example.  Let . The lines p'[1, 0, 5], q'[–3, 10, –11], and r'[3, –5, 22] are the respective images of the lines p[1, 2, 3], q[–1, 3, 1], and r[2, –1, 5]. Note that (1/2)p' = pA^–1, (1/6)q' = qA^–1, and (1/3)r' = rA^–1. The equations have three different values for k for the three lines, k = 1/2, 1/6, and 1/3, respectively.

Exercise 3.36. (a) Verify the above example.  

Find the inverse of matrix A,

 

Then the images of the lines are

 

 

 

 

(b) Find the matrix of an affine transformation that maps p[1, 2, 3], q[–1, 3, 1], and r[2, –1, 5] to p'[1, 0, 2], q'[–1, 5, –8], and r'[2, –5, 13], respectively.  (Hint. Need to solve a system of nine equations and nine variables. You may use a calculator or computer to solve the system.)

We need to find the values for the components of matrix

 

We apply Proposition 3.6, where both sides have been multiplied by the inverse, to the lines p, q, r, and their images:

 

 

 

These equations give us a system of nine equations in nine unknowns that we write in the following matrix equation:

 

Solving this system by the use of a computer, we obtain the matrix for the affine transformation of the Euclidean plane that maps p[1, 2, 3], q[–1, 3, 1], and r[2, –1, 5] to p'[1, 0, 2], q'[–1, 5, –8], and r'[2, –5, 13], respectively:

 and k₁ = k₂ = k₃ = 1.