Adv Alegbra HW Solutions 128

Adv Alegbra HW Solutions 128 - 128 4.14 Let k be a eld and...

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128 4.14 Let k be a fi eld, and let k n have the usual inner product. Prove that if v = a 1 e 1 + · · · + a n e n , then a i = (v, e i ) for all i . Solution. Prove that ( e i , e j ) = δ i j for all i , j , where δ i j is the Kronecker delta. 4.15 If f ( x ) = c 0 + c 1 x + · · · + c m x m k [ x ] and if A Mat n ( k ) , de fi ne f ( A ) = c 0 I + c 1 A + · · · + c m A m Mat n ( k ). Prove that there is some nonzero f ( x ) k [ x ] with f ( A ) = 0. Solution. Given A , then I , A , A 2 , . . . , A m 2 is a linearly dependent list because dim ( Mat m ( k )) = m 2 . 4.16 If U is a subspace of a vector space V over a fi eld k , then U is a subgroup of V (viewed as an additive abelian group). De fi ne a scalar multiplication on the cosets in the quotient group V / U by α(v + U ) = αv + U , where α k and v V . Prove that this is a well-de fi ned function that makes V / U into a vector space over k Solution. Let v + U V / U and let α be a scalar. If v + U = v + U , then v v U . Since U is a subspace, α(v v ) = αv αv U , and so αv + U = αv + U . Thus, scalar multiplication on V / U is well-de fi ned. 4.17 If V is a fi nite dimensional vector space and U is a subspace, prove that dim ( U ) + dim ( V / U ) = dim ( V ). Conclude that dim ( V / U ) = dim (
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