PROBLEM SET #1

DUE SEPTEMBER 8

1. Find 2 distinct bases for ℚ^{2}.

2. Find all 3 bases of 𝔽_{2}^{2}.

3. Are the vectors (1, 1, 1), (0, −1, 2), (2, 1, 3), and (4, 0, 11) linearly independent? If not, find a dependence relation.

4. Let a, b ∈ 𝔽 and let v⃗, w⃗ ∈ V. Show that

(a + b)(v⃗ + w⃗) = av⃗ + aw⃗ + bv⃗ + bw⃗.

5. Let C(ℝ) be the collection of all continuous functions from ℝ to itself. Let C_{+} denote the subset of “even” functions:

C_{+} = {f|f(−x) = f(x)},

and let C_{−} denote the “odd” functions:

C_{−} = {f|f(−x) = −f(x)}.

Show that both are subspaces of C(ℝ). Show also that C(ℝ) = C_{+} ⊕ C_{−}.

6. Let V be the vector space of all infinite sequences of real numbers. Show that the following are subspaces:

(1) ℓ_{1}, the subset of absolutely convergent sequences:

ℓ_{1} = {(a_{1}, …)| ∑ |a_{n}| < ∞}.

(2) z_{0}, the subset of eventually zero sequences.

(3) c_{0}, the subset of sequences which converge to zero.

(4) c, the subset of sequence which converge.

(5) ℓ_{∞}, the subset of bounded sequences:

ℓ_{∞} = {(a_{1}, …)| sup |a_{n}| < ∞}.