week4 - Math 115A Week 4 Textbook sections 2.3-2.4 Topics...

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Unformatted text preview: Math 115A - Week 4 Textbook sections: 2.3-2.4 Topics covered: • A quick review of matrices • Co-ordinate matrices and composition • Matrices as linear transformations • Invertible linear transformations (isomorphisms) • Isomorphic vector spaces * * * * * A quick review of matrices • An m × n matrix is a collection of mn scalars, organized into m rows and n columns: A = A 11 A 12 . .. A 1 n A 21 A 22 . .. A 2 n . . . A m 1 A m 2 . .. A mn . If A is a matrix, then A jk refers to the scalar entry in the j th row and k th column. Thus if A := 1 2 3 4 then A 11 = 1, A 12 = 2, A 21 = 3, and A 22 = 4. • (The word “matrix” is late Latin for “womb”; it is the same root as maternal or matrimony. The idea being that a matrix is a receptacle for holding numbers. Thus the title of the recent Hollywood movie “the Matrix” is a play on words). 1 • A special example of a matrix is the n × n identity matrix I n , defined by I n := 1 0 . .. 0 1 . .. . . . 0 0 . .. 1 or equivalently that ( I n ) jk := 1 when j = k and ( I n ) jk := 0 when j 6 = k . • If A and B are two m × n matrices, the sum A + B is another m × n matrix, defined by adding each component separately, for instance ( A + B ) 11 := A 11 + B 11 and more generally ( A + B ) jk := A jk + B jk . If A and B have different shapes, then A + B is left undefined. • The scalar product cA of a scalar c and a matrix A is defined by mul- tiplying each component of the matrix by c : ( cA ) jk := cA jk . • If A is an m × n matrix, and B is an l × m matrix, then the matrix product BA is an l × n matrix, whose co-ordinates are given by the formula ( BA ) jk = B j 1 A 1 k + B j 2 A 2 k + . .. + B jm A mk = m X i =1 B ji A ik . Thus for instance if A := A 11 A 12 A 21 A 22 and B := B 11 B 12 B 21 B 22 2 then ( BA ) 11 = B 11 A 11 + B 12 A 21 ; ( BA ) 12 = B 11 A 12 + B 12 A 22 ( BA ) 21 = B 21 A 11 + B 22 A 21 ; ( BA ) 22 = B 21 A 12 + B 22 A 22 and so BA = B 11 A 11 + B 12 A 21 B 11 A 12 + B 12 A 22 B 21 A 11 + B 22 A 21 B 21 A 12 + B 22 A 22 or in other words B 11 B 12 B 21 B 22 A 11 A 12 A 21 A 22 = B 11 A 11 + B 12 A 21 B 11 A 12 + B 12 A 22 B 21 A 11 + B 22 A 21 B 21 A 12 + B 22 A 22 . If the number of columns of B does not equal the number of rows of A , then BA is left undefined. Thus for instance it is possible for BA to be defined while AB remains undefined. • This matrix multiplication rule may seem strange, but we will explain why it is natural below. • It is an easy exercise to show that if A is an m × n matrix, then I m A = A and AI n = A . Thus the matrices I m and I n are multiplicative identities, assuming that the shapes of all the matrices are such that matrix multiplication is defined....
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This note was uploaded on 09/22/2011 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Fall '10 term at UCLA.

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week4 - Math 115A Week 4 Textbook sections 2.3-2.4 Topics...

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