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Adv Alegbra HW Solutions 125

Adv Alegbra HW Solutions 125 - 125(ii If PF(k F(k denotes...

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125 (ii) If PF ( k ) F ( k ) denotes the family of all polynomial functions a α n a n + · · · + α 1 a + α 0 , 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 { 0 } 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 { 0 } = 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 { 0 } and V . Assume V = { 0 } (otherwise dim ( V ) = 0 and we are done), and choose v V with v = 0. If u V , then v = V = v, u (for neither subspace is { 0 } ), so that v, u is a linearly dependent list. Therefore, dim ( V ) = 1. 4.4 Prove, in the presence of all the other axioms in the de fi nition of vector space, that the commutative law for vector addition is redundant; that is, if V
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