Answer: Let T : n n be an invertible linear

Chapter 2, Problem 38E

(choose chapter or problem)

Problem 38E

Let T : ℝn→ ℝn be an invertible linear transformation, and let S and U be functions from ℝn into ℝn such that

Show that

This will show that T has a unique inverse, as asserted in Theorem 9. [Hint: Given any vin ℝn, we can write v = T (x) for some x. Why? Compute S (v) and U (v).]

Theorem 9:

Let T : ℝn→ ℝn be a linear transformation and let A be the standard matrix for T . Then T is invertible if and only if A is an invertible matrix. In that case, the linear transformation S given by S (x) = A–1 is the unique function satisfying equations (1) and (2).