Algebra II MATH 251 Instructor: Dr. E. Goren.

Assignment 2

To be submitted by January 24, 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. Consider V := ℝ[t]_{n}, the vector space of polynomials of degree < n with real coefficients. Let

r_{1} < r_{2} < ⋯ < r_{n}

be any real numbers. Show that for every i there exists a unique polynomial f_{i} in V that vanishes at all the r_{j} except for r_{i} where it obtains the value 1. Give an explicit formula for f_{i}. Show that

f_{1}, f_{2}, …, f_{n}

comprise a basis for V.

6. Let B = {(1, 1), (1, 5)} and C = {(2, 1), (1, −1)} be bases of ℝ^{2}. Find the change of basis 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}.

Bonus question ( = 20%). Let 𝔽 be a finite field of q elements. Let V = 𝔽^{n} and let C be a code (= a subspace) of dimension k, hence having q^{k} elements. Let d be the minimal Hamming weight of a non zero element of C. Prove that

d ≤ n − k + 1.

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