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Unformatted text preview: 124 Exercises for Chapter 4
4.1 True or false with reasons.
(i) If k is a ﬁeld, then the subset E of all all polynomials of odd degree
is a subspace of k [x ].
Solution. True.
(ii) If A and B are n × n matrices over a ﬁeld k , and if the homogeneous system Ax = 0 has a nontrivial solution, then the homogeneous system ( B A)x = 0 has a nontrivial solution.
Solution. True.
(iii) If A and B are n × n matrices over a ﬁeld k , and if the homogeneous system Ax = 0 has a nontrivial solution, then the homogeneous system ( AB )x = 0 has a nontrivial solution.
Solution. False.
(iv) If v1 , v2 , v3 , v4 spans a vector space V , then dim(V ) = 4.
Solution. False.
(v) If k is a ﬁeld, then the list 1, x , x 2 , . . . , x 100 is linearly independent in k [x ].
Solution. True.
(vi) There is a linearly independent list of 4 matrices in Mat2 (R).
Solution. True.
(vii) There is a linearly independent list of 5 matrices in Mat2 (R).
Solution. False.
(viii) [Q( E 2π i /5 ) : Q] = 5.
Solution. False.
(ix) There is an inner product on R2 with (v, v) = 0 for some nonzero
v ∈ R2 .
Solution. True.
(x) The set of all f : R → R with f (1) = 0 is a subspace of F (R).
Solution. True.
4.2 (i) If f : k → k is a function, where k is a ﬁeld, and if α ∈ k , deﬁne
a new function α f : k → k by a → α f (a ). Prove that with this
deﬁnition of scalar multiplication, the ring F (k ) of all functions
on k is a vector space over k .
Solution. Straightforward veriﬁcation of the axioms. ...
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 Fall '11
 KeithCornell

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