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lect136_7_w09

# lect136_7_w09 - Friday January 16 Lecture 7 Standard matrix...

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Friday January 16 Lecture 7 : Standard matrix of a transformation . (Refers to section 3.3 ) Expectations : 1. Find the standard matrix induced by a linear transformation. Properties of linear transformations . 7.1 Definition Let { e 1 , e 2 , ..., e n } be a set of vectors in R n where e i is the vector containing only zeros except for the i th entry, which is a 1. We call this set of vectors the standard basis of R n . 7.2 Theorem Let T be a function mapping vectors from R n into R m . If T is a linear transformation then there exists a unique n x m matrix A such that T ( x ) = A x for all x in R n . In this case, A = [ c 1 c 2 ... c n ] where each column vector c i = T ( e i ) , i = 1 to n . Proof: Suppose T is a linear transformation. We are required to show that there exists a matrix A such T ( x ) = A x for all x in R n . Suppose x = ( x 1 , x 2 , x 3 , ..., x n ) in R n . Then x = x 1 e 1 + x 2 e 2 + ... + x n e n . For each i let c i = T ( e i ). Then T ( x ) = T ( x 1 e 1 + x 2 e 2 + ... + x n e n ) = x 1 T ( e 1 ) + x 2 T ( e 1 ) + ... + x n T ( e n ). = x 1 c 1 + x 2 c 2 + ... + x n c n . If A is the matrix [ c 1 c 2 ... c n ] whose columns are the vectors c i , then T ( x ) = A x . Hence the matrix A sends the vectors x in R n to precisely the same vectors as T does.

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