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
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.

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