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Unformatted text preview: MA 265 LECTURE NOTES: WEDNESDAY, FEBRUARY 20 Subspaces Definition. Let ( V, , ) be a real vector space. We say that a subset W V is a subspace of V if (1) W 6 = , and (2) ( W, , ) is a real vector space. We spend the rest of the lecture giving examples and discussing relevant results. Determining Subspaces. Let ( V, , ) be a real vector space, and W V be a subset. We give a method to determine when W is a subspace. We will show that W is a subspace of V if and only if (i) W is nonempty: W 6 = . (ii) W is closed under : If u , v W then u v W . (iii) W is closed under : If c R and u W then c u W . Hence in order to check that a subset is a subspace, it suffices to check that W is nonempty, closed under vector addition, and closed under scalar multiplication. We explain why this is true. First assume that W is a subspace of V . Then ( W, , ) is a real vector space. This means u v W for all u , v W so that (ii) holds; and c u W for all c R and u W so that (iii) holds. Since W this shows that (i) holds. Now assume that properties (i), (ii), and (iii) hold. We must show that (1) W 6 = , and (2) ( W, , ) is a real vector space. That is, if u , v , w W and c, d R then we must show that the following properties are valid: ( Commutativity: ) u v = v u ....
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- Spring '10