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Consider, for example, IR2 and d2. The statement is true, however, if the vector spaces have the same ground field. (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.

Xn_1,t} is a basis of V. ,xn_1,t} is also a basis of V. Show also that x2 E Kers(x1). Now show that if f : V -* V is the (unique) linear transformation such that f(xl) = x2, f(x2) = xl + x2, f(t) = t, f(xi) = xi (i 54 1,2) then f is an isomorphism that does not satisfy (*). Conclude from these observations that we must have n = 2. Suppose now that F has more than two elements and let A E F be such that A # 0,1. If there exists t 0 such that t E Ker S(t) observe that {t} is a basis of Ker S(t) and extend this to a basis It, z} of V.

31 Prove that if A is a normal matrix and g(X) is any polynomial then g(A) is normal. 32 If A and B are real symmetric matrices prove that A + iB is normal if and only if A, B commute. 33 Let A be a real skew-symmetric n x n matrix. Show that det(-A) _ (-1)1 det A and deduce thst if n is odd then det A = 0. Show also that every quadratic form xtAx is identically zero. Prove that the non-zero eigenvalues of A are of the form l i where µ E IR. If x = y + iz where y, z E IRn is an eigenvector associated with the eigenvalue iµ, show that Ay = -juz and Az = µy.

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A Necessary and Sufficient Condition for the Existence of a Stieltjes Integral (1917)(en)(5s) by Bliss G.A.


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