Exercise 2.6.15. Suppose that is a linear transformation from to itself, or more generally from any vector space to itself. Show that is also a linear transformation.

Answer: If is a linear transformation from some vector space to itself then by rule 2V on page 123 the composition of two linear transformations is also a linear transformation, again from to itself. So is a linear transformation if is.

Note that the assumption that maps from to itself is critical. If were a linear transformation from to a different vector space then it would not make sense to apply again to the result since is in not .

Another slightly longer proof that does not rely on rule 2V:

For clarity we use function notation, so that is the result of applying to a vector in . We also take to mean . In other words, is a composition of with itself. Since maps from to itself does also.

Since is a linear transformation we have for any and in and any two scalars and . Combining this with the definition of we have

Since we see that is also a linear transformation.

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.