Subspaces of polynomials with degree <= 2

[Inirit]

Homework Statement

Which of these subsets of P2 are subspaces of P2? Find a basis for those that are subspaces.

(Only one part)
{p(t): p(0) = 2}

Homework Equations

The Attempt at a Solution

So, I know the answer is that it's not a subspace via back of the book, but I don't understand why. Supposedly it's because it doesn't contain p(t) = 0 for all t, but I don't see why that's true. If p(0) = 2, then the equation has to look like 2 + bt + ct^2. If b,c = -1, then p(1) = 0, therefore there is a way for p(t) = 0.

Maybe I am just not understanding something. Admittedly, I feel really lost with all this.

 

[micromass]

Yes, there might be a t such that p(t)=0. But we need p(t)=0 for ALL POSSIBLE t.

In your example, we have the poly 2-t-t^2. Then p(1)=0, but that's not good enough. We want p(t)=0 for all t, not just 1.

 

[Inirit]

So... the neutral element of P2 has to equal 0 for all t? Therefore, a,b,c would have to all equal 0 to make that true, but that can't be the case because a has to equal 2, meaning it doesn't contain the neutral element and isn't a subspace of P2... right?

If that's the case, then it makes a bit more sense to me now.

 

[micromass]

Yes, this is correct!

 

[Inirit]

Awesome, thanks a lot for the help.