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Unformatted text preview: 125 (ii) If PF ( k ) ⊆ F ( k ) denotes the family of all polynomial functions a 7→ α n a n + ··· + α 1 a + α , prove that PF ( k ) is a subspace of F ( k ) . Solution. P ( k ) is closed under addition and scalar multiplication 4.3 Prove that dim ( V ) ≤ 1 if and only if the only subspaces of a vector space V are { } and V itself. Solution. Assume that dim ( V ) = 1 (the result is obvious if dim ( V ) = 0). Let e be a basis of V , let { } = S ⊆ V , and let s = 0 lie in S . The list e , s must be linearly dependent, so that there are scalars α, β , not both 0, with α e + β s = 0. Now α = 0, lest β s = 0 and β = 0 (for s = 0). Hence, e = α − 1 β s ∈ S , forcing V ⊆ S , and so S = V . Conversely, suppose the only subspaces of V are { } and V . Assume V = { } (otherwise dim ( V ) = 0 and we are done), and choose v ∈ V with v = 0. If u ∈ V , then h v i = V = h v, u i (for neither subspace is { } ), so that v, u is a linearly dependent list. Therefore, dim ( V...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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