Linear Algebra homework (needed it in 5 hrs)

HOMEWORK 4

DUE MONDAY APRIL 30TH AT 2PM

Your homework should be written on standard-sized paper, and loose sheets

must be stapled together.

(1) Let V be a vector space over a field F. (a) Prove that if A ⊆ V is a spanning set for V , then so is B for any B ⊇ A. (b) Prove that if S ⊆ V is a linearly independent set over F, then so is T for

any T ⊆ S.

(2) (a) Find a basis for the subspace

U = {(a, b, c, d) | 2a = 3c and b = −d}⊆ R4.

(b) Extend your basis from (a) to a basis for R4. (c) Find a subspace W of R4 such that U ⊕W = R4.

(3) (a) Find a basis for the subspace

U = {p(x) ∈ P3(R) | p(2) = 0}⊆ P3(R).

[Note: p(2) means “evaluate the polynomial at 2”, i.e. sub in x = 2.]

(b) Extend your basis from (a) to a basis for P3(R). (c) Find a subspace W of P3(R) such that U ⊕W = P3(R).

(4) Let V be a vector space over a field F, and suppose that there are elements v1, v2, v3 ∈ V such that V = spanF(v1 + v2, v3 −v2, v1, v2). (a) Show that {v1 + v2, v3 −v2, v1, v2} cannot be a basis for V . (b) What is the maximum dimension that V could have? Give an example to

show that this maximum can be attained.

(5) Consider the following three functions in RR, which we view as a vector space with the usual addition and scalar multiplication over R:

f : R −→ R : x 7→ x2, g : R −→ R : x 7→ ex, h : R −→ R : x 7→ |x|.

Prove that {f, g, h} is a set of linearly independent elements in RR.

[Hint: write out the definition of linear independence and the zero element of

RR explicitly. What does it mean for two functions to be equal?]

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