l7 - Math 20F Linear Algebra Lecture 7: 2.1 Matrix algebra....

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Unformatted text preview: Math 20F Linear Algebra Lecture 7: 2.1 Matrix algebra. An m n matrix A is a collection of numbers (called entries) [ a ij ] 1 i m, 1 j n j th column A = a 11 ... a 1 j ... a 1 n . . . a i 1 ... a ij ... a in . . . a m 1 ... a mj ... a mn i th row we often write it in terms of column vectors: A = [ a 1 a n ] , a j = a 1 j . . . a mj or just A = [ a ij ]. We will use the notation ( A ) ij to denote the ( i,j ) th entry a ij . Addition : Two m n matrices A and B can be summed together by summing their entries. ( A + B ) ij = ( A ) ij + ( B ) ij Example: 1 2 3 4 + 1- 3 2 = 1 + 0 2 + 1 3- 3 4 + 2 = 1 3 0 6 Subtraction can be defined in a similar way. Scalar multiplication : An m n matrix A can be multiplied by a scalar by multiplying all the entries by . ( A ) ij = ( A ) ij Example: 3 1 2 3 4 5 6 = 3 1 3 2 3 3 3 4 3 5 3 6 = 3 6 9 12 15 18 The zero matrix : There is a zero matrix of each shape. For example the 4 4 zero matrix is 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . Laws of arithmetic: If A , B and C are m n matrices, 0 is the zero m n matrix and r,s as scalars, then we can the usual laws of arithmetic hold for matrices, for...
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This note was uploaded on 04/29/2008 for the course MATH 20F taught by Professor Buss during the Spring '03 term at UCSD.

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l7 - Math 20F Linear Algebra Lecture 7: 2.1 Matrix algebra....

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