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Unformatted text preview: Lecture 30 6 Manifolds 6.1 Canonical Submersion and Canonical Immersion Theo rems As part of todays homework, you are to prove the canonical submersion and im mersion theorems for linear maps. We begin todays lecture by stating these two theorems. Let A : R n R m be a linear map, and let [ a ij ] be its associated matrix. We have the transpose map A t : R m R n with the associated matrix [ a ji ]. Definition 6.1. Let k < n . Define the canonical submersion map and the canonical immersion map as follows: Canonical submersion: : R n R k , ( x 1 , . . . , x n ) ( x 1 , . . . , x k ) . (6.1) Canonical immersion: : R k R n , ( x 1 , . . . , x k ) ( x 1 , . . . , x k , , . . . , 0) . (6.2) Canonical Submersion Thoerem. Let A : R n R k be a linear map, and suppose that A is onto. Then there exists a bijective linear map B : R n R n such that A B = . Proof Hint: Show that there exists a basis v 1 , . . . , v n of R n such that Av i = e i , i = 1 , . . . , k , (the standard basis of R k ) and Av i = 0 for all i > k . Then let B : R n R n be the linear map Be i = v i , i = 1 , . . . , n , where e i , . . . , e n is the standard basis of R n ....
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
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