Wilfred Kaplan's Advanced Calculus PDF

By Wilfred Kaplan

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Rl, . . , x,. Both points of view will be important in this chapter. In either case we call A = ( a i j )the coeficient matrix of the set of equations. The numbers y , , . . , y, can be considered as the entries in a column vector y = col ( y l . . , y,). The numbers x , , . . YI,. . , X,). Let A = ( a i j )and B = ( b i j ) be matrices of the same size, both m x n. Then one defines the sum A B to be the m x n matrix C = ( c i j )such that cij = a,, b,, for all i and j ; that is, one adds two matrices by adding corresponding entries.

Hence by Rules (d) and (g) we must have k _( n. Let us suppose k < n. Then we can express e l , . . , en as linear combinations of vl, . . , vk-say, Abl = e l , . . , Ab,, = en, where A is the n x k matrix whose column vectors are vl, . . Accordingly, A B = I, where B is the k x n matrix whose column vectors are b,, . . , b,. 9). Hence we have a contradiction. and we must have k = n. Thus Rule (i) is proved. The basis e l , . . 105) is called the standard basis of V " . For n = 3 this is the familiar basis i, j, k.

D) Prove from the rcsult of (c) that A and B havc equal traces. [ I ] is not similar to a diagonal matrix. ] 7. Prove that the matrix A = Chapter 1 Vectors and Matrices 8. Prove the following: a) Every square matrix is similar to itself. b) If A is similar to B and B is similar to C, then A is similar to C. 12 THETRANSPOSE Let A = (a;,) be an m x n matrix. We denote by A' the n x m matrix B = ( b i j )such that hi, = a,; for i = 1 , . . , n , j = 1, . . , m . Thus B = A' is obtained from A by interchanging rows and columns.

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Advanced Calculus by Wilfred Kaplan


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