**Math 55a: Honors Abstract Algebra** *Homework Assignment #5 (30 Septemer 2016): Linear Algebra V: “Eigenstuff”* *(with a prelude on exact sequences and more duality)* The terms “proper value”, “characteristic value”, “secular value”, and “latent- value” or “latent root” are sometimes used [for “eigenvalue”] by other authors. The latter term is due to Sylvester [Collected Papers III, 562-4] because such numbers are “latent in a somewhat similar sense as vapour may be said to be latent in water or smoke in a tobacco-leaf.” We will not adhere to his termi- nology. - N. Dunford and J.T. Schwartz: Linear Operators, Part I, pages 606-7. A bit about exact sequences: 1. i)Suppose0→V1 →V2 →V3 →···→Vn →0isanexactsequenceoflinear transformations between vector spaces all of which are finite dimensional. Prove that Pni=1(−1)i dim Vi = 0. ii) Given positive integers di (i = 1, . . . , n) such that Pni=1(−1)idi = 0, must there exist an exact sequence as in (i) such that dim Vi = di for each i? The next two questions explore further aspects of duality. For problem 2, vectors v1, . . . , vN in an n-dimensional vector space V are said to be “in general linear posi- tion” if every choice of n vectors vi1,...,vin with i1 < i2 < ··· < iN yields a basis for V. For example, this condition is satisfied by vi = (1,xi) ∈ F2 for any pairwise distinct xi ∈ F (even though they are quite special in that any three points are collinear). Moregenerally(1,xi,x2i,...,xdi)worksinFd+1,againassumingthexi are pairwise distinct.
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