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Unformatted text preview: 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: firstname.lastname@example.org. This is a handout for Sections 13, 15, and 16. Any typos and errors are solely the responsibility of the author. 1 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 ) = A~x ). 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...
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