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week01 - Math 205 Spring 2006 B Dodson New text...

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Math 205, Spring 2006 B. Dodson New text - Peterson-Sochacki ———- 1. Course Info 2. Week 1 Homework: 1.1 (1st half), 1.2 Intro Due Thurs! ———- An m × n matrix A is an array with m horizontal rows; n vertical columns i, j th entry a i,j in the i th row, and j th column. row vector or row n -vector, a, is a 1 × n matrix, just one row. column vector or column n -vector, b, is a n × 1 matrix, just one column. ———- Example. Give the rows and columns of A = 2 10 6 5 - 1 3 . What are the entries a 1 , 2 , a 2 , 1 , a 3 , 1 , a 1 , 3 ? The matrix sum, A + B, is defined only when A and B have the same shape; and then the i, j th entry of A + B is a i,j + b i,j , the sum of the i, j th entries of A and B.
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2 The scalar multiple of the matrix A by the scalar (number!) c is the matrix with the same shape as A, but with i, j th entry of ca i,j . Matrix multiplication: 1. row n -vector by column m -vector, only when n = m, is the number a 1 b 1 + a 2 b 2 + · · · + a n b n , where the a i are the entries of a and the b i are the entries of b. 2. m × n matrix A by p -column vector b, only when n = p, is the m -column vector with i th entry ( i th row of A ) · b. ——– Problem 3.2.11 Multiply A = - 1 2 4 7 5 - 4 by c = 5 - 1 . Solution: Ac = - 1 2 4 7 5 - 4 5 - 1 = ( - 1)(5) + 2( - 1) 4(5) + 7( - 1) 5(5) + ( - 4)( - 1) = - 7 13 29 .
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