Solution to Exercise 3.58.
Upon the education of the people of this country the fate of this country depends.
—Benjamin Disraeli (1804–1881)

Exercise 3.58. Find a matrix of the translation that maps the line l[2, 3, –1] to m[2, 3, 5]. Is the matrix unique?

 

We apply Proposition 3.6; there is a nonzero real number k such that kmA = l where A is the matrix of an affine translation.

 

 

implies

 

Hence, k = 1 and 2a +3b + 5 = –1. For one example of a matrix that translates line l to line m, let a = 0 and b = –2;

 

Note the matrix A is not unique. A general form of a matrix that translate line l to line m is

 where a is a real number.