Is W={P(x) | P(x) in P3 and P(0) = 1} a subspace of P3? The question is as follows:
Is W={P(x) | P(x) in P3 and P(0) = 1} a subspace of P3? If so find a basis and dimension.
I am not sure how to answer this question in full.
So far I have the following work done for this question:
P(0) = 1
Any element of the vector space P3 is of the form:
a + bx + cx^2 + dx^3
Sub in x=1
a + b + c + d = 0
Take any two vectors v and u to be:
v = a1 + b1x + c1x^2 + d1x^3
u = a2 + b2x + c2x^2 + d2x^3
Now check to see if they are closed under addition:
r = v+u = (a1 + a2) + (b1 + b2)x + (c1 + c2)x^2 + (d1 + d2)x^3
r = a3 + b3x + c3x^2 + d3x^3
So its closed under addition?
Now check to see if they are closed under scalar multiplication:
c*v = cv = ca1 + cb1x + cc1x^2 + cd1x^3
c*u = cu = ca2 + cb2x + cc2x^2 + cd2x^3
So its closed under scalar multiplication?

The set $W$ is not a subspace of $P_3$. For instance, the constant polynomial $1$ belongs to $W$, but $1+1$ doesn't.
