Linear Algebra, 5th Edition

2.3 Composition of Linear Transformations and Matrix Multiplication

In Section 2.2, we learned how to associate a matrix with a linear transformation in such a way that both sums and scalar multiples of matrices are associated with the corresponding sums and scalar multiples of the transformations. The question now arises as to how the matrix representation of a composite of linear transformations is related to the matrix representation of each of the associated linear transformations. The attempt to answer this question leads to a definition of matrix multiplication. We use the more convenient notation of UT rather than $\text{U}\circ \text{T}$ for the composite of linear transformations U and T. (See Appendix B.)

Our first result shows that the composite of linear ...