Page 681 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The three successive row operations can be performed by multiplying on the left successively by

Inverting these matrices and using 5 yields

Reading from right to left and noting that

it follows that the effect of multiplying by A is equivalent to
1. shearing by a factor of 2 in the x-direction,
2. then expanding by a factor of 2 in the y-direction,
3. then reflecting about the x-axis,
4. then shearing by a factor of 3 in the y-direction.

The proofs for parts of the following theorem are discussed in the exercises.
THEOREM 9.2.2

Images of Lines
If is multiplication by an invertible matrix, then

(a) The image of a straight line is a straight line.
(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.
(d) The image of the line segment joining points P and Q is the line segment joining the images of P and Q.