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

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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 + 0 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 λ by multiplying all the entries by λ . ( λA ) ij = λ ( 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 example A + B = B + A, A + 0 = A, A - A = 0 , ( A + B ) + C = A + ( B + C ) ( rs ) A = r ( sA ) , r ( A + B ) = rA + rB etc.

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