Use the Exercise 20 to prove parts (b) and (c) of Theorem 2.
A line in the plane has an equation of the form Ax + By + C = 0, where A and B are not both zero. Use the method of Example 8 to show that the image of this line under multiplication by the invertible matrix

has the equation A€²x + B€²y + C = 0, where
A€² = (dA - cB) / (ad - bc) and B€² = (-bA + aB) / (ad - bc)
Theorem 2
If T: R2 †’ R2 is multiplication by an invertible matrix, then
(b) The image of a straight line through the origin is a straight line through the origin.
(c) The images of parallel straight lines are parallel straight lines.

2