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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 3-space, R 3 ). That is, there is a function between R and R 2 which is one-to-one 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 1-dimensional, the plane 2-dimensional, and 3-space 3-d. 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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