New PDF release: Advanced Calculus

By Wilfred Kaplan

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1 (n times). For each m and n we define the m x n zero matrix One sometimes denotes the matrix by Om, to indicate the size-that is, the number of rows and columns. In general, two matrices A = (ai,) and B = (bij) are said to be equal, A = B, when A and B have the same size and aij = b;; for all i and j . A 1 x n matrix A is formed of one row: A = ( a l l , . . , a,,). We call such a matrix a row vector. 48), each of the successive rows forms a row vector. We often denote a row vector by a boldface symbol: u, v , .

Or, in handwriting, by an arrow). 49) has the row vectors u l = (2, 3 , 5 ) andu2 = (1,2,3). Chapter 1 Vectors and Matrices Similarly, an m x 1 matrix A is formed of one column: We call such a matrix a column vector. For typographical reasons we sometimes denote this matrix by col ( a l l ,. . , a m l )or even by ( a l l ,. . , a m l ) ,if the context makes clear that a column vector is intended. We also denote column vectors by boldface letters: u, v, . . 49) has the column vectors vl = co1(1,4) and v2 = col(2, 3).

Then as remarked above, also det B # 0. so that B also has an inverse B - ' , and BB-' = I . We can now write BA = BAI = B A B B ~ '= B ( A B ) B ~= ' BIB-' = BB--' = I Therefore, also, BA = I . Furthermore, if AC = I , then This shows that the inverse,of A is unique. Furthermore, if CA = I , then C=CI=CAB=IB= B. r B satisfies either one of these two equations, then B must satisjj die other equation. and B = A - ' . The inverse satisfies several additional rules: Here A and D are assumed to be nonsingular n x n matrices.

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Advanced Calculus by Wilfred Kaplan


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