Berkeley MATH 54 - The inverse of a Linear Transformation (9 pages)

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The inverse of a Linear Transformation



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The inverse of a Linear Transformation

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Lecture Notes


Pages:
9
School:
University of California, Berkeley
Course:
Math 54 - Linear Algebra and Differential Equations

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The inverse of a Linear Transformation September 10 2007 Recall Let S and T be sets A mapping or function from S to T is a rule which assigns to every element s of S a welldefined element f s of T The set S is called the source or domain of f The set T is called the target or codomain of f A function f S T is said to be invertible if for every t in T there is a unique s in S such that f s t A closer look at f s t f is said to be injective one to one if for every t there is at most one s such that f s t f is said to be surjective onto if for every t there is at least one s such that f s t f is said to be bijective if for every t there is exactly one s such that f s t Inverse functions Theorem Let f S T be a function Then the following conditions are equivalent f is surjective and injective f is bijective There exists a function g T S such that g f s s for all s and f g t t for all t This g is called the inverse of the function f The inverse of a linear transformation Theorem Let A be an n x m matrix Then TA Rm Rn is invertible if and only if n m rank A If this is the case its inverse TA 1 is also linear Proof Let r be the rank of A If r m then there are m r free variables Hence the equation TA x 0 has infinitely many solutions and TA could not be injective Thus if TA is injective m r hence m r Conversely if m r TA is injective If r n then there is some y such that the equation TA x y is inconsistent Hence TA will not be surjective Hence if TA is surjective n r hence n r Conversely if n r TA is surjective Thus if TA is bijective n r m Conclusion If A is an n x m matrix of rank r 1 TA is injective if and only if m r 2 TA is surjective if and only if n r Corollary If n m then the following are equivalent 1 TA is injective i e r m 2 TA is surjective i e r n 3 TA is bijective Computing TA 1 If TA is invertible then TA 1 T A 1 another linear transformation where A 1 is computed as follows Form the matrix A In Put A In in reduced row echelon form Then rref A In In A 1 See



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