![]() ![]() We use the notation S V to indicate that S is a subspace of V and S < V to indicate that S is a proper subspace of V, that is, S V but S V. If the answer to both of these questions is yes, then \(U\) is a vector space. Subspaces Definition 1.5.1 A subspace of a vector space V is a subset S of V that is a vector space in its own right under the operations obtained by restricting the operations of V to S. ![]() More information on subspaces can be found in Gilbert Strangs excellent. ![]() In other words, the set of vectors is closed under addition v Cw and multiplication cv (and dw). EXAMPLE 1 The Space R1 Consider the follows subset of R: R1 (v 1 v 2 v 3 :::) 2R v n 0 for all but nitely many n: Note that R1is closed under addition and scalar multiplication, and is therefore a linear. The following example discusses one such subspace.
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