1. ## Subspace- Urgent

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.

2. ## yes

p(2) = 0 => a*p(2)=0
p(2) = 0 and q(2) = 0 => p(2)+q(2) = 0
p(x) = 0 is in H

p(2) = 0 => a*p(2)=0
p(2) = 0 and q(2) = 0 => p(2)+q(2) = 0
p(x) = 0 is in H
Where in the world do you get q(t)? That is, q(2)?

4. Notice p is a polynomial..but I don't know how you get q...

5. ## .

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.