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Unformatted text preview: Spanning, linear dependence, dimension In the crudest possible measure of these things, the real line R and the plane R 2 have the same size (and so does 3space, R 3 ). That is, there is a function between R and R 2 which is onetoone and onto. But everybody knows that there is a definite sense in which the plane is “bigger” than the line, and Eu clidean space is bigger still. The line is 1dimensional, the plane 2dimensional, and 3space 3d. One of the purposes of these notes is to make this idea precise. In what follows, we assume we have fixed a field F and a vector space V over F . We start, of course, with a DEFINITION (LINEAR COMBINATION). Suppose that S ⊆ V and ~v ∈ V . (So S is a set of vectors from V .) We say that ~v is a linear combination of the vectors in S if there are ~v 1 ,..., ~v k in S and scalars α 1 ,..., α k such that ~v = α 1 ~v 1 + ··· + α k ~v k . For a simple example, suppose that F = R , V = R 3 and S = 3 1 5 ,  7 2 . Then 6 9 12 is a linear combination of the vectors in S because 6 9 12 = 2 3 1 5 + ( 1)  7 2 . But 6 9 11 is not a linear combination of the vectors in S because we cannot solve the system α 1 3 1 5 + α 2  7 2 = 6 9 11 . (Try it.) (This example brings up a small matter of language. In case S = { ~v 1 ,...,~v k } is a fairly small finite set — and it usually will be, for us — and ~v is a linear combo of the vectors in S , we will often just say that ~v is a linear combination of ~v 1 ,..., ~v k . So it is correct (and standard) to say that 6 9 12 is a linear combination of 3 1 5 and  7 2 . However, we do want to leave open the possibility that S is infinite. But even if S is infinite, a linear combination involves only finitely many of the vectors from S .) For a slightly more sophisticated example, again suppose that F = R , but V = C ( R ), the space of continuous functions on the real numbers. Then cos2 x is a linear combination of 1 and sin 2 x . This follows immediately from the well 1 known fact that cos2 x = 1 2sin 2 x (for all x ). [Please note that as usual in this context, 1 does not denote the number but the constant function.] Here’s a closely related notion: DEFINITION (DEPENDENCE RELATION). Suppose that ~v 1 , ..., ~v k are dis tinct vectors in V and there are scalars α 1 ,..., α k which are not all zero such that α 1 ~v 1 + ··· + α k ~v k = ~ 0. In such a case we call the expression α 1 ~v 1 + ··· + α k ~v k = ~ a (nontrivial) dependence relation ....
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This note was uploaded on 10/03/2010 for the course MATH 223 taught by Professor Loveys during the Spring '07 term at McGill.
 Spring '07
 Loveys
 Linear Algebra, Algebra

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