m235notes7 - Notes 7: The Inverse of a Matrix We show how...

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Notes 7: The Inverse of a Matrix We show how to solve certain systems of linear equations. The ingredients of this approach are two. First is the idea of the inverse function to a function. The second idea is Gauss elimination and the ideas around Gauss elimination such as rank. These ideas will be used often. Definition 1. Let F : S -→ T, G : T -→ S be two functions. Assume that G F = Id S and F G = Id T , that is, for all s S,G F ( s ) = s and for all t T,F G ( t ) = t . We then say that F is the inverse function to G and G is the inverse function to F . Why are we interested in inverse functions? Answer: If we are given an equation F ( x ) = a to solve and if G is the the inverse to F we can apply it to the equation to get G F ( x ) = G ( a ) x = G ( a ) . Example 1 . Given f : R -→ R x 7→ mx m R ,m 6 = 0 , then the inverse to f is the function g ( x ) = m - 1 x so the solution to the equation mx = a is x = g ( a ) = m - 1 a. Example 2 . The function f : R -→ R x 7→ x 2 does not have an inverse because f only takes on positive values. If we start with x = - 1 then there is no way to define g ( - 1) while staying in the real numbers so that f g ( - 1) = - 1 . Let R 0 denote the set of non-negative real numbers. Example
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m235notes7 - Notes 7: The Inverse of a Matrix We show how...

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