Introduction to Abstract Algebra, Solutions Manual, 4th Edition

6.1 Vector Spaces

1.

a. No. 0∉ U.

c. No. 0 ∉ U and not closed under addition.

2.

a. Yes. This is because (2f)(x) = 2f(x) and (f + g)(x) = f(x) + g(x) for all f, g F[x].

c. No. 0 ∉ U.

3.

4.

a. The inclusion ⊇ is clear. Since

and similarly for and , 89 we have also proved the inclusion ⊆.

5. We have , so adding −(av) to both sides gives Similarly yields

7.

a. Dependent. (1, 2, 3) + 2(4, 0, 1) + 3(2, 1, 0) = (0, 0, 0).

c. Independent. a(x² + 1) + b(x + 1) + cx = 0 gives a = 0, b + c = 0, a + b = 0; so a = b = c = 0.

9.

a. If then ; . Thus 2ab = 0 and a² + 2b² = 3c². If a = 0, ...