Adv Alegbra HW Solutions 124

Adv Alegbra HW Solutions 124 - 124 Exercises for Chapter 4...

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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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