09/30/04

In theorem nine we will now see that if S exists then it must be a linear transformation and is unique. S can also be called the inverse of T and can be written as T-1.

Theorem Nine:

Let T: R^n à R^n 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 this case, the linear transformation S given by S(x) = A-1 x is the unique function satisfying (1) and (2).

Proof:

Suppose T is invertible then (2) shows that T is onto R^n, for if b is in R^n  and x = S(b) the T (x) =  T(s(b)) = b so each b is in the range of T. Thus A is invertible by theorem 8 letter (i).

Conversly, suppose A is invertible and let S(x) = A-1 x. Then S S is a linear transformation

S(t(x)) = S(Ax) = A-1 (Ax) = x  

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This site was last updated 09/30/04