- #1

- 13

- 0

## Homework Statement

Let [tex]P_4(\mathbb{R})[/tex] be the vector space of real polynomials of degree less than or equal to 4.

Show that {[tex]{f \in P_4(\mathbb{R}):f(0)=f(1)=0}[/tex]}

defines a subspace of V, and find a basis for this subspace.

## The Attempt at a Solution

Since [tex]P_4(\mathbb{R})[/tex] is spanned by (1,z,...,z

^{4}), I think that this subspace will be spanned by a list/set of 4 elements - since f(0)=0 means there is no constant term. And obviously, by the definition of basis, these elements will have to be linearly independent.

I really don't know how I would represent a basis for this subspace; I am inclined to consider x(x-1) as an element of degree 2. But it just seems wrong to me to fix that as an element in the basis: (z, z

^{2}-z, z

^{3}, z

^{4}).

But are not the roots of polynomials determined by their coefficients? Seems to me that I need to find some real coefficients so that f(0)=f(1)=0 is guaranteed.

So not only am I uncertain about how to express a basis of this subspace, but I am not even sure of this approach.

This subspace is defined by polynomials whose roots are 0, 1, and beyond that, any real number. As a subspace is closed under scalar multiplication, multiplying the elements of the basis (which I have yet to obtain) must produce elements with roots that include 0 and 1. But given how the roots are determined by the coefficients, I really don't know how to fix this relationship.

I would greatly appreciate any help and direction, since I'm probably confusing a lot of things at once.

Thanks in advance.