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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.
- Fall '04