5 - MATHEMATICS 54 Professor Constantin Teleman 9/10/09...

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MATHEMATICS 54 Professor Constantin Teleman 9/10/09 Lecture 5 ASUC Lecture Notes Online is the only authorized note-taking service at UC Berkeley. Do not share, copy or illegally distribute (electronically or otherwise) these notes. Our student-run program depends on your individual subscription for its continued existence. These notes are copyrighted by the University of California and are for your personal use only. D O N O T C O P Y Sharing or copying these notes is illegal and could end note taking for this course. ANNOUNCEMENTS Today we will spend 20 minutes on Schubert bases and then wrap up what we know about solutions of linear system. LECTURE Method: Write them as rows into a matrix and bring it to a reduced echelon form. Then the non- zero rows of the reduced row echelon form the Schubert basis of Span { r r 1 ,..., r r k }. Observations: The row space (span of the rows) of any matrix is the row space of its echelon form. Example: Plane in R 3 , x+y+z=0, plane P. r r 1 r r 2 10 1 01 1 The longest initial string is 0’s having a zero above the pivot and we can get longest initial string of 0’s and leading 1. Theorem: The Schubert basis of a subspace of R n is unique. Consequence (Corollary): Two subspace of agree if and only if they have the same Schubert basis. Back to example: P= Span 1 2 1 1 1 0 1 21 1 1 1 1 1 r v s 1 0 1
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This note was uploaded on 12/01/2009 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at University of California, Berkeley.

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5 - MATHEMATICS 54 Professor Constantin Teleman 9/10/09...

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