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Unformatted text preview: NOTES ON SECTION 4 . 3 MA265, SECTIONS 41, 52 Some key words Subspace Zero subspace subset parametric equation Closure linear combination null space 1. Subspaces A subspace of a vector space is a subcollection W of the objects of a vector space ( V, , ) which themselves form a vector space when you use these operations (that is, ( W, , ) is itself a vector space). Definition 1. A subspace of a vector space ( V, , ) is a subset W V such that (1) For every u,v W , u v is also in W (2) For every u W and scalar r R , r u is also in W . We said that ( W, , ) should be a vector space, and in fact the above definition implies that this is the case. That is, one can deduce by going through the axioms one by one (see Theorem 4.3 in the text): Theorem 2. If W is a subspace (as defined above), then ( W, , ) is itslef a vector space. Example 3. Zero subspace (ex.1 in text) Example 4. Polynomials of degree less than or equal to 4 in the space of all polynomials. Or P n [ x ] P m [ x ] for n < m . Example 5. The set of points in R 2 x y satisfying the condition (1) x is NOT a subspace (2) x + y is NOT a subspace (3) x = 0 IS a subspace...
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