Linear algebra problem subspaces?

Question

Which of the following subsets of P2 are subspaces of P2?

A. { p(t) | p'(t) + 3 p(t) + 1 =0 }
B. { p(t) | p'(t) is constant }
C. { p(t) | p(8)=2 }
D. { p(t) | p(5)=0 }
E. { p(t) | p(-t)=p(t) for all t }
F. { p(t) | p'(4)=p(1) }

Answer 1

A subspace must contain the zero polynomial, so this rules out (A) and (C).

Otherwise a subset is a subspace if all linear combinations of any two members of the subset is also in the subset.

(B) is a subspace, because if u' and v' are constant, so is (a*u+b*v)' = a*u'+b*v'.

(D) is a subspace because if u(5) = 0 and v(5) = 0, then a*u(5)+b*v(5) = 0 too.

(E) is the zero subspace

(F) also a subspace, since if u,v are in S = {p | p'(4)=p(1)} iff:
u'(4) - u(1) = v'(4) - v(1) = 0
The derivative is linear, so:
(a*u + b*v)'(4) - (a*u + b*v)(1)
= a*u'(4) - a*u(1) + b*v'(4) - b*v(1)
= 0
So, (a*u + b*v) are in S for all a,b.