Chapter 3-5 Notes

Chapter 3-5 Notes - HANDOUT MATH 294 Vadim Zipunnikov1...

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H ANDOUT M ATH 294 Vadim Zipunnikov 1 C HAPTER 3 C OORDINATES IN A SUBSPACE OF R n Definition 3.4.1 Consider a basis B of a subspace V of R n , consisting of vectors v 1 , v 2 , . . . , v m . Any vector x in V can be written uniquely as x = c 1 v 1 + c 2 v 2 + · · · + c m v m . The scalars c 1 , c 2 , . . . , c m are called B -coordinates of x , and the vector [ c 1 , c 2 , . . . , c m ] is called the B -coordinate vector of x , denoted by [ x ] B . Note that x = S [ x ] B , where S = | | | v 1 v 2 · · · v m | | | T HE MATRIX OF A LINEAR TRANSFORMATION Definition 3.4.2 Consider a linear transformation T : R n -→ R n , and a basis B of R n . The n × n matrix B that transforms [ x ] B into [ T ( x )] B is called the B -matrix of T: [ T ( x )] B = B [ x ] B for all x in R n . T HE COLUMNS OF THE MATRIX OF A LINEAR TRANSFORMATION Fact 3.4.3 Consider a linear transformation T : R n -→ R n , and a basis B of R n consisting of vectors v 1 , v 2 , . . . , v n . Then, the B -matrix is B = [ T ( v 1 )] B , [ T ( v 2 )] B , . . . , [ T ( v n )] B ; that is, the columns of B are the B -coordinate vectors of T ( v 1 ) , T ( v 2 ) , . . . , T ( v n ) . S TANDARD MATRIX VERSUS B -MATRIX OF A LINEAR TRANSFORMATION Fact 3.4.4 1 Mailing address: Malott Hall 128, Cornell University, Ithaca, 14850, NY, USA. E-mail: [email protected] This is a handout for Sections 13, 15, and 16. Any typos and errors are solely the responsibility of the author. 1

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Consider a linear transformation T : R n -→ R n , and a basis B of R n consisting of vectors v 1 , v 2 , . . . , v n . Let B be the B -matrix of T and let A be standard matrix of T (such that T ( x ) = Ax ). Then AS = SB, B = S - 1 AS, and A = SBS - 1 , where S = | | | v 1 v 2 · · · v n | | | S IMILAR MATRICES Definition 3.4.5 Consider two n × n matrices A and B. We say that A is similar to B if there is an invertible matrix S such that AS = SB, or B = S - 1 AS S IMILARITY IS AN EQUIVALENCE RELATION Fact 3.4.6 a. An n
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