lecture30

lecture30 - Lecture 30 6 Manifolds 6.1 Canonical Submersion...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.

Page1 / 3

lecture30 - Lecture 30 6 Manifolds 6.1 Canonical Submersion...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online