Algebra Through Practice: Volume 4, Linear Algebra: A by T. S. Blyth, E. F. Robertson

By T. S. Blyth, E. F. Robertson

Problem-solving is an artwork important to figuring out and talent in arithmetic. With this sequence of books, the authors have supplied a range of labored examples, issues of entire recommendations and try papers designed for use with or rather than typical textbooks on algebra. For the ease of the reader, a key explaining how the current books can be used along side the various significant textbooks is integrated. every one quantity is split into sections that commence with a few notes on notation and conditions. the vast majority of the cloth is aimed toward the scholars of general skill yet a few sections include more difficult difficulties. by means of operating throughout the books, the scholar will achieve a deeper knowing of the basic innovations concerned, and perform within the formula, and so resolution, of different difficulties. Books later within the sequence disguise fabric at a extra complicated point than the sooner titles, even supposing each one is, inside its personal limits, self-contained.

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Example text

Xiii) False. 0 0 is a counter-example. (xiv) False. For example, 1 [0 0 1 0][1 0]- [0 1 4][0 0] (xv) True. (xvi) True. (xvii) False. Take, for example, f, g : IR" -* IR' given by f (x, y) (0, 0) and g(x, y) = (x, y). Relative to the standard basis of IR' we see Linear algebra Book 4 that f is represented by the zero matrix and g is represented by the identity matrix; and there is no invertible matrix P such that P-142P = 0. (xviii) True. (xix) False. The transformation t is non-singular (an isomorphism), but r1 21 1 2J is singular.

Prove that (1) if det A = 1 and n is odd, or if det A = -1 and n is even, then 1 is an eigenvalue of A; (2) if det A = -1 then -1 is an eigenvalue of A. 28 If A is a skew-symmetric matrix and g(X) is a polynomial such that g(A) = 0, prove that g(-A) = 0. Deduce that the minimum polynomial of A contains only terms of even degree. Deduce that if A is skew-symmetric and f (X), g(X) are polynomials whose terms are respectively odd and even then f (A), g(A) are respectively skew-symmetric and symmetric.

V = Ker t ® Im t holds in cases (i) and (ii), but not in case (iii); for in case (iii) we have that (1, 1, 1) belongs to both Kert and Imt. 4 If s o t = idv then s is surjective, hence bijective (since V is of finite dimension). Then t = s-1 and so t o s = idv. Suppose that W is t-invariant, so that t(W) C_ W. Since t is an isomorphism we must have dimt(W) = dim W and so t(W) = W. Hence W = s[t(W)] = s(W) and W is s-invariant. The result is false for infinite-dimensional spaces. For example, consider the real vector space IR[X] of polynomials over IR.

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