Linear Algebra, 5th Edition

2.4 Invertibility and Isomorphisms

The concept of invertibility is introduced quite early in the study of functions. Fortunately, many of the intrinsic properties of functions are shared by their inverses. For example, in calculus we learn that the properties of being continuous or differentiable are generally retained by the inverse functions. We see in this section (Theorem 2.17) that the inverse of a linear transformation is also linear. This result greatly aids us in the study of inverses of matrices. As one might expect from Section 2.3, the inverse of the left-multiplication transformation ${\text{L}}_A$ (when it exists) can be used to determine properties of the inverse of the matrix A.

In the remainder of this section, we apply many of the results ...