lecture 18, August 22, 2018.pdf

Its easy to check practice problem that for each 1 i

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It’s easy to check (practice problem) that for each 1 i n , A e i = a i and B e i = b i . But then the columns of A and B are identical, hence A = B .
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Conclusion: given a linear transformation f : R n R m , we find the standard matrix A for f by finding the vectors f ( e 1 ) , f ( e 2 ) , . . . , f ( e n ) and arranging them into the matrix A = f ( e 1 ) f ( e 2 ) · · · f ( e n ) .
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Example Consider the linear function f : R 2 R 3 defined by f x 1 x 2 = 2 x 1 + x 2 x 1 - 3 x 2 - x 1 + x 2 . We compute f 1 0 = 2 1 - 1 , f 0 1 = 1 - 3 1 , and therefore the corresponding standard matrix is A = 2 1 1 - 3 - 1 1 .
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As promised, we now give the proof that a matrix has only one reduced row echelon form. Theorem The reduced echelon form of an m × n matrix A is unique.
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Note: This proof is an adaptation of the one appearing in the article: The Reduced Row Echelon Form of a Matrix is Unique , by Thomas Yuster. It appeared the March 1984 issue of Mathematics Magazine. Proof. The proof goes by induction on n . For the case n = 1, there are only two m × 1 matrices that satisfy the requirements of reduced row echelon form: they are 0 = 0 · · · 0 T , and e T 1 = 1 0 · · · 0 T . However, there can be no sequence E 1 , . . . , E N of m × m elementary matrices such that E N · · · E 1 0 = e T 1 , simply because E 0 = 0 for all elementary matrices E . Therefore we have proved the base case.
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For the induction step, suppose we are given an m × ( n + 1) matrix A . Let a 1 , . . . , a n +1 be the columns of A . Set A 0 = a 1 . . . a n . Suppose there are two distinct reduced row echelon forms for A . Let’s call them B and C . By definition, there exist two sequences of m × m elementary matrices, say , E 1 , . . . E N , and F 1 , . . . F M , such that B = E N · · · E 1 A = E N · · · E 1 A 0 E N · · · E 1 a n +1 , and C = F M · · · F 1 A = F M · · · F 1 A 0 F M · · · F 1 a n +1 .
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Now, the key observation to make is that, because B and C are both in reduced row echelon form, so are the matrices E N · · · E 1 A 0 and F M · · · F 1 A 0 (if you delete the last column of matrix of an m × ( n
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Christopher Reinemann
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