Let T: Rⁿ → Rⁿ be an invertible linear transformation, and let S and U be functions from Rⁿ into Rⁿ such that S (T(x)) = x and U (T(x)) = x for all x in R". Show that U(v) = S(v) for all v in R". This will show that I has a unique inverse, as asserted in Theorem 9.

Data from in Theorem 9

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If A is a 2 x 2 matrix, the area of the parallelogram determined by the columns of A is det Al. If A is a 3 x 3 matrix, the volume of the parallelepiped determined by the columns of A is det A.