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

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).