Problem 1 Show that

S = {(x,y) ∈R2 : x2 > y}

is not a subspace of R2 by finding vectors u,v S such that u + v /S.

Problem 2 Let V be the vector space consisting of all differentiable functions f : (0,1) → (0,1). Show that

by checking the three criteria

The constant function zero is in S.

If f,g S, then (f + g) ∈ S also.

If f S and c is a constant, then cf S also.

Problem 3 Show that

S = {A M2×2 : A2 is the zero matrix}

is not a subspace of M2×2.

Problem 4 Find three 3 × 2 matrices which span the subspace

null(A)}

Problem 5 Let

S = {f P3 : f(1) = f(2)}

Determine the general form of a polynomial in S. Find three cubic polynomials which span S.