Exercise 2.6.11. Consider the vector space of functions in for which . From the previous exercise we can express any such function as where and are basis vectors for W.

Suppose at we have and . Find and such that fulfills these initial conditions. This can be considered a linear transformation from the vector space of vectors into the vector space . Find the matrix corresponding to this transformation assuming the basic vectors and for and and for .

Answer: At we have

We also have

so that at we have

From our initial conditions and at we then have

Adding the two equations we have or

Subtracting the second equation from the first we have or

The linear transformation from into can be represented by the following matrix:

so that if is in we have

where is in .

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.