They play a central role in fast fourier transforms and in

Question

Prob. 1. Let A₁ = 

a) Prove or disprove: (i) Y ∈ S_P(A₁, A₂, A₃);
                             (ii) Z ∈ S_p(A₁, A₂, A₃)

b) FIND a basis for S_p(A₁, A₂, A₃).

Prob. 2. Let V = P₃ be the space of cubic polynomials. Let e₁ (t) = t³; e₂(t) = t²(1 - t); e₃(t) = t(1 - t)²; e₄(t) = (1-t)³.

SHOW that every v ∈ P₃ has a unique representation v = _i=1Σ⁴ c_ie_i(t)

Prob. 3. Let B_N = set of all linear combinations of e^ikt k = -N,........,0.......N. (Such linear combinations form a set of "band limited" functions. They play a central role in Fast Fourier Transforms and in digital signal processing, among other things.) It is easy to see that Bay is a vector space of functions. Assume that this is so.

a) SHOW {1, cost, cos Nt, sint ....... sin Nt} is a basis for B_N

b) What is Dim(B_N)?
Prob. 4. V = S_p (1, cost, cos 2t, cos3t, cos 4t) is a vector space.

a) SHOW {1, cos t, cos²t, cos³t, cos⁴t is a basis for V
b) What is Dim (V)?

Prob. 5. Let V be the vector space of all 2 x 2 matrices over the real numbers. PROVE that V has dimension 4 by exhibiting a basis for V which has four elements.

Prob. 6. Let V and W both be subspaces of a given vector space. PROVE or DISPROVE: Their intersection V ∩ W(= {x : x ∈ V and x ∈ W}) is also a subspace. (Remember, there are at least three facts you have to consider.)