By Wilfred Kaplan

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U . v = v . u . VII. ou = 0. IX. ( u + v ) . w = u . w + v . w . X. ( a u ) . v = a ( u . v ) . X I . u . u 2 0. XII. u . u = 0 if and only if u = 0. 102) A set of k vectors v l ,. . , vk of V n is said to be linearly independent if an equation + + + + + + + + can hold only if c l = . . = ck = 0. 103) does hold with not all c's equal to 0, then the vectors are said to be linearly dependent. Remarks. We observe that for every n x k matrix A , Acol(ci, . . , c ~ ) = c I v I + . . 104) where v i , .

A2 the imaginary pariof A. + + + - d ) If A is a square matrix, then (A)'= (A') and, if A is nonsingular, then ( A p ' )= ( A ) - ' . e) A is a real matrix if and only if A = A . 11. Let A be a real square matrix, let h be real, and let Av = hv for a nonzero complexcolumn vector v. Show that Au = hu for a real nonzero vector u , so that h is an eigenvalue of A considered as a real matrix. [Hint: Let v = p i q , where p and q are real and not both zero. Show, with the aid of the results of Problem 10, that A p = hp and Aq = hq and hence that u can be chosen as one of p, q .

A,, = det A b) Concludc from the result of (a) that det A = det B C) Provethati! + . . + A , = a l l + . . + a , , , , . T h e n u n l b e r a l ~ + . . +a,,, iscalledthe trace of A. 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.

### Advanced Calculus by Wilfred Kaplan

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