cal6 - Chapter Six Linear Functions and Matrices 6.1...

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6.1 Chapter Six Linear Functions and Matrices 6.1 Matrices Suppose f : R R n p be a linear function. Let e e e 1 2 , , , K n be the coordinate vectors for R n . For any x R n , we have x e e e = + + + x x x n n 1 1 2 2 K . Thus f f x x x x f x f x f n n n n ( ) ( ) ( ) ( ) ( ) x e e e e e e = + + + = + + + 1 1 2 2 1 1 2 2 K K . Meditate on this; it says that a linear function is entirely determined by its values f f f n ( ), ( , ( ) e e e 1 2 K . Specifically, suppose f a a a f a a a f a a a p p n n n pn ( ) ( , , , ( ) ( , , , ( ) ( , , , ). e e e 1 11 21 1 2 12 22 2 1 2 = = = K K M K Then f a x a x a x a x a x a x a x a x a x n n n n p p pn n ( ) ( , , , x = + + + + + + + + + 11 1 12 2 1 21 1 22 2 2 1 1 2 2 K K K K The numbers a ij thus tell us everything about the linear function f . . To avoid labeling these numbers, we arrange them in a rectangular array, called a matrix :
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6.2 a a a a a a a a a n n p p pn 11 12 1 21 22 2 1 2 K K M M K The matrix is said to represent the linear function f . For example, suppose f : R R 2 3 is given by the receipt f x x x x x x x x ( , ) ( , , ) 1 2 1 2 1 2 1 2 2 5 3 2 = - + - . Then f f ( ) ( , ) ( , , ) e 1 10 213 = = , and f f ( ) ( , ) ( , , ) e 2 01 15 2 = = - - . The matrix representing f is thus 2 1 3 1 5 2 - - Given the matrix of a linear function, we can use the matrix to compute f ( ) x for any x . This calculation is systematized by introducing an arithmetic of matrices. First, we need some jargon. For the matrix A a a a a a a a a a n n p p pn = 11 12 1 21 22 2 1 2 K K M M K ,
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6.3 the matrices [ ] a a a i i in 1 2 , , , K are called rows of A , and the matrices a a a j j pj 1 2 M are called columns of A . Thus A has p rows and n columns; the size of A is said to be p n × A vector in R n can be displayed as a matrix in the obvious way, either as a 1 × n matrix, in which case the matrix is called a row vector , or as a n × 1 matrix, called a column vector . Thus the matrix representation of f is simply the matrix whose columns are the column vectors f f f n ( ), ( , ( ) e e e 1 1 K Example Suppose f : R R 3 2 is defined by f x x x x x x x x x ( , , ) ( , ) 1 2 3 1 2 3 1 2 3 2 3 2 5 = - + - + - .
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This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

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cal6 - Chapter Six Linear Functions and Matrices 6.1...

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