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Math20_Fall2011_Basis - Harvard University Math 20 Fall...

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Unformatted text preview: Harvard University, Math 20 Fall 2011, Instructor: Rachel Epstein 1 Notes on Vector Spaces, Bases, and Dimension Definition 0.1. A vector space is a set of vectors V ⊂ R n that satisfies the following properties: 1. If ~v , ~w are in V , then ~v + ~w is in V . 2. If ~v is in V , then c~v is in V for all real numbers c . 1 Notice that ~ 0 is in every vector space. In fact, every vector space is the span of a finite set of vectors. (Recall that the span of a set of vectors is the set of all linear combinations of the vectors.) For every n , R n is a vector space. Another example is the set of all 2-vectors ( x,y ) where x = y . A non-example: the set of all vectors in R 2 that lie on the x-axis or the y-axis. As you can see, (1 , 0) and (0 , 1) are both in that set, but their sum (1 , 1) is not. An important example of a vector space is the set of solutions to any homo- geneous system of equations....
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