Exercise 2.6.13. Suppose that is a linear transformation from the – plane to itself. If a transformation exists such that show that is also linear. Also show that if is the matrix representing then the matrix representing must be .
Answer: Consider the expression for any and . We know from the definition of that
Since is a linear transformation we then have
and using the definition of we have and so that
From our final result and initial expression we have
for all and . So if is a linear transformation and exists then is a linear transformation also.
Now suppose that is the matrix representing and is the matrix representing . By 2V on page 123 the composition of and is itself a linear transformation, with represented by the product matrix and represented by the product matrix .
But since for all we know that the compositions of and are equivalent to the identity transformation represented by the identity matrix . We therefore have and which implies that .
So if is the matrix representing then is the matrix representing .
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.