This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 Bcoordinates of ~x , and the vector [ c 1 , c 2 , . . . , c m ] is called the Bcoordinate 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 Bmatrix 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 Bmatrix is B = [ T ( ~v 1 )] B , [ T ( ~v 2 )] B , . . . , [ T ( ~v n )] B ; that is, the columns of B are the Bcoordinate vectors of T ( ~v 1 ) , T ( ~v 2 ) , . . . , T ( ~v n ) . S TANDARD MATRIX VERSUS BMATRIX OF A LINEAR TRANSFORMATION Fact 3.4.4 1 Mailing address: Malott Hall 128, Cornell University, Ithaca, 14850, NY, USA. Email: vvz2@cornell.edu. 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 Bmatrix 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...
View Full
Document
 Fall '05
 HUI
 Math, Vectors

Click to edit the document details