does this set of polynomials span the space P3(R)?

Question

Does the linearly independent set {2x^3-1, x^2+x} span P3(R), the vector space of real valued polynomials with degree less than or equal to 3?

For that matter, does the set consisting of a single polynomial: {x^3+x^2+x+1} span P3(R)? Since it is a linear combination of the standard basis vectors for P3(R) = {x^3, x^2, x, 1}??

Answer 1

As others have said, you need 4 vectors, so "no" is the answer.

You can see that these are not sufficient by trying to take the given vectors and seeing if you can use them in a linear combination to get 1, x, x^2 or x^3.



From the first set, any linear combination will result in a polynomial where x and x^2 have the same coefficient. So you can get 2x^2 + 2x but you can't get x^2 or x by itself.



With the single vector, all you ever get are polynomials where all terms have the same coefficient on every power of x.



With 4 linearly independent vectors, you can solve for the combination that gives you 1, x, x^2 or x^3. You would have 4 independent equations in 4 unknowns (being the scalars that you'd apply to the vectors), and you could solve that for the unique combination to give you the standard basis.

Answer 2

If P3(R) means the space of polynomials of degree 3 or less, then it is 4 dimensional so no set of less than 4 vectors can span.

Answer 3

I think a set needs 4 lindep vectors in order to span P3(R), since the space has dimension 4. Since both of your sets are short by a few vectors, they merely span subspaces.