Spanning, linear dependence, dimension
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Spanning, linear dependence, dimension

Course: MATH 223, Spring 2010

School: McGill

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Spanning, linear dependence, dimension In the crudest possible measure of these things, the real line R and the plane R2 have the same size (and so does 3-space, R3 ). That is, there is a function between R and R2 which is one-to-one and onto. But everybody knows that there is a denite sense in which the plane is bigger than the line, and Euclidean space is bigger still. The line is 1-dimensional, the plane...

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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 1 ,..., k such that ~v = 1 ~v 1 + + k ~v k .

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