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An introduction to the theory of linear spaces by Shilov G.

By Shilov G.

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What can be said about the relation between a set of vectors and its span? There are three easy statements on the most abstract (and therefore most shallow) level, namely that (1) every set is a subset of its span, (2) if a set E is a subset of a set F, then the span of E is a subset of the span of F, and (3) the span of the span of a set is the same as the span of the set. It is often convenient to have a symbol to denote \span", and one possible symbol is (which is intended to be reminiscent of the ordinary set-theoretic symbol for union).

25 25. Spans Do linear combinations of more than two vectors make sense? Sure. If, for instance, x, y, and z are three vectors in R3 , or, for that matter, in R2 or in R20 (see Problem 22) and if α, β, and γ are scalars, then the vector αx + βy + γz is a linear combination of the set {x, y, z}. Linear combinations of sets of four vectors, such as {x1 , x2 , x3 , x4 }, are defined similarly as vectors of the form α 1 x1 + α 2 x2 + α 3 x3 + α 4 x4 (where α1 , α2 , α3 , and α4 are scalars, of course), and the same sort of definition is used for linear combinations of any finite set of vectors.

Of monomials is a total set. For the space O, the empty set is total. 35 VECTORS The good vector spaces in linear algebra, the easiest ones to work with and the ones that the subject is rooted in, are the ones that have a finite total set; vector spaces like that are called finite-dimensional. The space P2 is finite-dimensional, but (see Problem 27) the space P is not. The first natural question about finite-dimensional vector spaces sounds deceptively simple: is every subspace of a finite-dimensional vector space finite-dimensional?

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