Kunze Linear Algebra · Problem 3.2.9

Linear Transformations

Mathematics — Methods · one-sided-inverse-implies-invertible

Problem

Let

T

be a linear operator on the finite-dimensional space

V

. Suppose there is a linear operator

U

V

such that

TU = I

. Prove that

T

is invertible and

U = T^{-1}

. Give an example which shows that this is false when

V

is not finite-dimensional. (Hint: Let

T = D

, the differentiation operator on the space of polynomial functions.)