Set of degree 2 polynomials a subspace

In summary, the conversation discusses which subsets of P2 are subspaces and finding a basis for those that are subspaces. It is determined that the subset (P(t)|p(0) = 2) is not a subspace because it does not contain the function f(t) = 0 for all t. The reasoning behind this is that f(0) is not equal to 2 when a=0, b=0, and c=0.
  • #1
wumple
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Homework Statement


Which of the subsets of P2 given in exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces.

(P(t)|p(0) = 2)


Homework Equations





The Attempt at a Solution


The solution manual says that this subset is not a subspace because it doesn't contain the function f(t) = 0 for all t. I thought the generic element is f(t) = a +bt + ct^2. Why doesn't the element with a = b = c = 0 count as a function f(t) = 0 for all t? I'm stumped.

Thanks!
 
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  • #2
Because f(0) isn't equal to 2 with a=0, b=0 and c=0. f(t) isn't in your set.
 
  • #3
Oh, duh. thanks!
 

1. What is a set of degree 2 polynomials?

A set of degree 2 polynomials is a collection of polynomials where the highest degree term is 2. This means that the variable in the polynomial is raised to the power of 2, and there can also be lower degree terms present.

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

In order for a set of degree 2 polynomials to be a subspace, it must satisfy three conditions: closure under addition, closure under scalar multiplication, and contain the zero vector (a polynomial with all coefficients equal to 0).

3. Can a set of degree 2 polynomials contain polynomials of different degrees?

No, a set of degree 2 polynomials can only contain polynomials where the highest degree term is 2. If there are polynomials of different degrees present, then it is not considered a set of degree 2 polynomials.

4. How many basis vectors are needed to span a set of degree 2 polynomials?

A set of degree 2 polynomials can be spanned by a basis of two polynomials. These basis polynomials are typically x^2 and x, but any two linearly independent degree 2 polynomials can also serve as a basis.

5. What is the dimension of a set of degree 2 polynomials?

The dimension of a set of degree 2 polynomials is 2, as it can be spanned by a basis of two polynomials. This means that any polynomial in the set can be written as a linear combination of these two basis polynomials.

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