Adv Alegbra HW Solutions 128

Adv Alegbra HW Solutions 128 - 128 4.14 Let k be a field,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 128 4.14 Let k be a field, and let k n have the usual inner product. Prove that if v = a1 e1 + · · · + an en , then ai = (v, ei ) for all i . Solution. Prove that (ei , e j ) = δi j for all i , j , where δi j is the Kronecker delta. 4.15 If f (x ) = c0 + c1 x + · · · + cm x m ∈ k [x ] and if A ∈ Matn (k ), define f ( A) = c0 I + c1 A + · · · + cm Am ∈ Matn (k ). Prove that there is some nonzero f (x ) ∈ k [x ] with f ( A) = 0. 2 Solution. Given A, then I , A , A2 , . . . , Am is a linearly dependent list 2. because dim(Matm (k )) = m 4.16 If U is a subspace of a vector space V over a field k , then U is a subgroup of V (viewed as an additive abelian group). Define 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-defined 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-defined. 4.17 If V is a finite dimensional vector space and U is a subspace, prove that dim(U ) + dim(V / U ) = dim(V ). Conclude that dim(V / U ) = dim(V ) − dim(U ). Solution. Choose a basis u 1 , . . . , u m of U , so that dim(U ) = m , and extend it to a basis of V by adjoining vectors v1 , . . . , vr . Show that v1 + U , . . . , vr + U is a basis of V / U , and conclude that dim(V / U ) = r = dim(V ) − m . 4.18 Let Ax = b be a linear system of equations, and let s be a solution. If U is the solution space of the homogeneous linear system Ax = 0, prove that every solution of Ax = b has a unique expression of the form s + u for u ∈ U . Conclude that the solution set of Ax = b is the coset s + U . Solution. Absent. 4.19 If U and W are subspaces of a vector space V , define U + W = {u + w : u ∈ U and w ∈ W }. (i) Prove that U + W is a subspace of V . Solution. Absent. ...
View Full Document

Ask a homework question - tutors are online