Review exercise 2.10. Given the set of all linear transformations from to define operations for scalar multiplication and vector addition that will make the set a vector space. What is the dimension of the resulting vector space?
Answer: Let and be linear transformations from to and let and be the by matrices corresponding to and respectively. If is a scalar define the scalar product to be the linear transformation represented by the matrix and define the vector sum to be the linear transformation represented by the matrix .
The set of linear transformations from to is a vector space under the operations thus defined; this follows from the fact that the set of x matrices is a vector space under those operations. The dimension of the space is .
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.