Review exercise 2.29. The following matrices

represent linear transformations in the – plane with and as a basis. Describe the effect of each transformation.

Answer: When the matrix is applied to the vector we obtain

When the matrix is applied to the vector we obtain

The effect of the transformation represented by is to reflect all vectors through the -axis.

When the matrix is applied to the vector we obtain

When the matrix is applied to the vector we obtain

When the matrix is applied to the vector we obtain

The transformation represented by is a shearing transformation that leaves all vectors on the -axis unchanged but changes the coordinates of all other vectors in proportion to their distance from the -axis.

When the matrix is applied to the vector we obtain

When the matrix is applied to the vector we obtain

The effect of the transformation represented by is to rotate all vectors through an angle of -90 degrees. (Or, in other words, clockwise through 90 degrees.)

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.