Points, vectors, linear independence and some introductory linear algebra
نویسندگان
چکیده
منابع مشابه
Some linear algebra
Before we can show how to apply the ring theory that we have developed to linear transformations of vector spaces we need to set up some results on polynomial factorizaton. Let F be a field and let x be an indeterminate. Set R = F [x]. Then as we have seen R is a P.I.D. so in particular a unique factorization domain. Noting that if f(x), g(x) ∈ F [x] then deg(f(x)g(x)) = deg(f(x)) + deg(g(x)) w...
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defined for all complex numbers λ, where I denotes the � × � identity matrix. It is not hard to see that a complex number λ is an eigenvalue of A if and only if χA(λ) = 0. We see by direct computation that χA is an �th-order polynomial. Therefore, A has precisely � eigenvalues, thanks to the fundamental theorem of algebra. We can write them as λ1� � � � � λ�, or sometimes more precisely as λ1(A...
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In this paper we deduce a lower bound for the rank of a family of p vectors in Rk (considered as a vector space over the rationals) from the existence of a sequence of linear forms on Rp, with integer coefficients, which are small at k points. This is a generalization to vectors of Nesterenko’s linear independence criterion (which corresponds to k = 1). It enables one to make use of some known ...
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ژورنال
عنوان ژورنال: Journal of Chemometrics
سال: 2016
ISSN: 0886-9383
DOI: 10.1002/cem.2802