Subspaces of polynomials with degree <= 2

In summary, the conversation discussed the subsets of P2 that are subspaces and finding a basis for them. One of the subsets, {p(t): p(0) = 2}, was determined to not be a subspace because it does not contain the neutral element p(t) = 0 for all t. The explanation for this was that in order to be a subspace, p(t) = 0 must be true for all possible values of t, not just one. This was illustrated through an example where p(1) = 0, but that is not enough to fulfill the requirement. It was then concluded that because a, b, and c cannot all equal 0 in the given subset, it does not
  • #1
Inirit
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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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
Yes, this is correct!
 
  • #5
Awesome, thanks a lot for the help.
 

1. What is a subspace of polynomials with degree <= 2?

A subspace of polynomials with degree <= 2 is a subset of the set of all polynomials with degree <= 2 that satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.

2. How do you determine if a set of polynomials with degree <= 2 is a subspace?

To determine if a set of polynomials with degree <= 2 is a subspace, you can check if it satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and containing the zero vector. If it satisfies all three properties, then it is a subspace.

3. What is the dimension of a subspace of polynomials with degree <= 2?

The dimension of a subspace of polynomials with degree <= 2 is 3, since it is a subset of the vector space of all polynomials with degree <= 2, which has a basis of three linearly independent polynomials.

4. How does a subspace of polynomials with degree <= 2 relate to linear independence?

A subspace of polynomials with degree <= 2 can be thought of as a set of linearly independent polynomials. This means that no polynomial in the set can be written as a linear combination of the other polynomials in the set. This property is important in determining the dimension and basis of the subspace.

5. Can a subspace of polynomials with degree <= 2 contain polynomials with degree > 2?

No, a subspace of polynomials with degree <= 2 can only contain polynomials with degree <= 2. This is because the set must be closed under addition and scalar multiplication, and adding or multiplying a polynomial with degree > 2 would result in a polynomial with degree > 2, which is not part of the subspace.

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