Algebra II 189-251B Instructor: Dr. E. Goren.

Assignment 2

To be submitted by January 26, 12:00

1. Let V be the set of all sequences of complex numbers (a_{0}, a_{1}, a_{2}, …) satisfying a_{n} = a_{n−1} + a_{n−2}, ∀n ≥ 2.

Show that V has a natural structure of a vector space over ℂ. Find its dimension and a basis.

2. Let V be an n-dimensional vector space over a field 𝔽. Let T = {t_{1}, …, t_{m}} ⊂ V be a linearly independent set. Let W = Span(T). Prove:

dim(W) = m.

3. Let W be a subspace of a vector space V of dimension n. Let {t_{1}, …, t_{m}} be a basis for W. Prove that there exist vectors {t_{m+1}, …, t_{n}} in V such that {t_{1}, …, t_{m}, t_{m+1}, …, t_{n}} is a basis for V.

4. Let V_{1}, V_{2} be finite dimensional vector spaces over a field 𝔽. Prove that dim(V_{1} ⊕ V_{2}) = dim(V_{1}) + dim(V_{2}).

5. Let 𝔽 be a field and a_{11}, …, a_{1n} be scalars in 𝔽. Let W_{1} be the set of solutions to the linear equation ∑_{j=1}^{n} a_{1j}x_{j} = 0, namely,

W_{1} = {(x_{1}, …, x_{n}) ∈ 𝔽^{n} : a_{11}x_{1} + ⋯ + a_{1n}x_{n} = 0}.

Note that W_{1} is a subspace of 𝔽^{n}. Prove:

(1) dim(W_{1}) ≥ n − 1.

(2) Assume we are given scalars a_{ij}, 1 ≤ i ≤ k, 1 ≤ j ≤ n. Assume also that k < n. Let W be

the solution set of the system of equations

a_{11}x_{1} + ⋯ + a_{1n}x_{n} = 0
a_{21}x_{1} + ⋯ + a_{2n}x_{n} = 0
⋮  ⋮
a_{k1}x_{1} + ⋯ + a_{kn}x_{n} = 0.

Prove that dim(W) ≥ n − k. (Suggestion: Argue by induction on k. Use dim(W_{1} + W_{2}) = dim(W_{1}) + dim(W_{2}) − dim(W_{1} ∩ W_{2})).

6. Let B = {(1, 1), (1, 5)} and C = {(2, 1), (1, −1)} be bases of ℝ^{2}. Find the change of bases matrices BM_{C} and _{C}M_{B} between the bases B and C. Let v = ( \begin{matrix}8 \\ 28 \end{matrix} ) with respect to the standard basis. Find [v]_{B} and [v]_{C}.

7. Let 𝔽 be a finite field with q elements.

(1) Show that the kernel of the ring homomorphism

ℤ ⟶ 𝔽

defined by n ↦ n · 1 = 1 + ⋯ + 1 (n times) is of the form pℤ for some prime p. Conclude that we may assume that 𝔽 ⊇ ℤ/pℤ for some prime p.

(2) Prove that 𝔽 is a vector space of finite dimension over ℤ/pℤ and if this dimension is n then 𝔽 has p^{n} elements^{1}.

1Note: at this point you’ve proven that every finite field has cardinality p^{n} for some prime p.