Unformatted text preview: a transformation T : F m → F n such that L A ◦ T is the identity. Then T = L B for some matrix B , and this is the matrix we seek as then I m = [ L A ◦ T ] = [ L A L B ] = [ L AB ] = AB . We construct T by prescribing its behavior on a basis for F m . Let { e 1 , . . . , e m } be the standard basis for F m . For each e i , there exists x i ∈ F n such that L A ( x i ) = e i since L A is onto. Deﬁne T by setting T ( e i ) = x i for each i . Then L A ( T ( e i )) = e i for every i , and so L A ◦ T is the identity on F m . This proves the claim and the result. 3.3.10 This statement is true. If the m × n coeﬃcient matrix A has rank m , then L A : F n → F m is onto. Thus (by previous hw problem) any equation Ax = b has a solution....
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 Winter '07
 Liu
 Math, Linear Algebra, Algebra, Trigraph, cx cy cz

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