Given H = {P $\displaystyle = p(t) \in P_3 : p(2) = 0$}
Is H a subspace of $\displaystyle P_3$?
Prove the 3 properties of subspace (0 in H, closed under add./mult.) or show an example that violates a property.
NOTE: P means vector.
p and q are arbitrary elements of H. What you are showing is that H is closed under addition and multiplication by a scalar and is non-empty ie.
p is an element of H and q is an element of H => p+q is an element of H
and
p is an element of H => a*p is an element of H for an arbitrary scalar 'a'
and
there exists p such that p is an element of H.
This is the very basics of subspaces, so I suggest you talk to your teacher if you don't recognise what I am saying.