# 8 - MATHEMATICS 54 Professor Constantin Teleman Lecture 8...

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Professor Constantin Teleman 9/22/09 Lecture 8 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. LECTURE Last time we have talked about determinants of square matrices n x n . We use: *Tests for invertibility (det 0). *Formula for the invertible matrix. *Formula for solutions to b x A if detA 0 (Cramer). Computation theorem: Row-reduction, determinant equals to the product of the pivots (if n pivots), row-column expansion, big formula (n! terms), and is theoretically useful. Properties: det (In) =1, det is multilinear in the columns of A. Anti-symmetric=skew-symmetric=alternating Det (A)=det ( ) T A , det (AB)=detA x detB Geometric meaning: det=(signed) volume of the slanted box defined by the n columns of A. Today we will talk about Chapter 4. We will cover vector spaces, subspaces, linear independence/spanning/bases, linear map, kernels, ranges, and coordinate changes (new formula). Definition: An abstract vector space is a set V of “vectors” of the following structures. Main example: n R , operations: addition of vectors, scalar multiplication. (“Every finite-dimensional vector space can be identified within n R ”) Vector space: “ n R without the standard basis”. Abstract vector space is a set of V of “vectors” with the following structure: VxV V ( RxV V These must satisfy a long list of properties (“axioms”). Same as the operations on n R ! such that

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8 - MATHEMATICS 54 Professor Constantin Teleman Lecture 8...

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