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 paintings primary to realizing and talent in arithmetic. With this sequence of books, the authors have supplied a variety of labored examples, issues of whole recommendations and try out papers designed for use with or rather than general textbooks on algebra. For the ease of the reader, a key explaining how the current books can be utilized along with the various significant textbooks is incorporated. every one quantity is split into sections that start with a few notes on notation and stipulations. the vast majority of the fabric is geared toward the scholars of commonplace skill yet a few sections comprise more difficult difficulties. by means of operating throughout the books, the coed will achieve a deeper realizing of the elemental techniques concerned, and perform within the formula, and so resolution, of alternative difficulties. Books later within the sequence hide 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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An(n - 1) 0 ... 1 1 1 1 1 0 0 1 2 3 0 1 0 0 0 n The eigenvalues of t are all 1. The characteristic polynomial is g(X) = (X - 1)n+1 Hence, by the Cayley-Hamilton theorem, g(t) = 0. The minimum poly- nomial of t is then m(X) = (X - 1)' for some r with 1 < r < n + 1. A simple check using the above matrix shows that (t - idv)" 54 0 for 1 < r < n. 22 Let A1, ... , An be the eigenvalues of t. Then A', . . =Azn =1. Consequently, at = ±1 for each i and hence the sum of the eigenvalues of t is an integer.

Then ac + bd = idv. Suppose now that v is an eigenvector of ab associated with the eigen- value 0. ) Let u = a(cv) and w = b(dv). Then since a, b, c commute we have bu = bacv = cabv = 0, and since a, b, d commute we have aw = abdv = dabv = 0. Also, u + w = (ac + bd)v = v since ac + bd = idv. 26 Ifu,vE CA then t(u)=Au and t(v) = Av and so t(au + bv) = at(u) + bt(v) = aAu + bAv = A(au + bv) and hence Ca is a subspace of V. Let v E CA. Then, since s and t commute, we have t(s(v)] = s[t(v)] = s(Av) = As(v) from which it follows that Ca is s-invariant.

X - iy] + +x2+y2 x+iy and a sum of squares is zero if and only if each summand is zero. Hence we see that Y = 0. The minimum polynomial of A cannot have repeated roots. For, if this were of the form m(X) = (X - a)2p(X) then from (A - aI)2p(A) = 0 we would have, by the above applied to each column of p(A) in turn, (A - aI)p(A) = 0 and m(X) would not be the minimum polynomial. 52 Solutions to Chapter 1 Thus the minimum polynomial has simple roots and so A is similar to a diagonal matrix. Suppose now that Ax = iax.

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