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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 welldeﬁned 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 welldeﬁned.
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. ...
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 Fall '11
 KeithCornell

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