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## Homework Statement

Consider the vector space F(R) = {f | f : R → R}, with the standard operations.

Recall that the zero of F(R) is the function that has the value 0 for all

x ∈ R:

Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have

the same value at x = −1 and x = 1.

Define functions g; h; j and k ∈ F[R] by

g(x) = 2x3 − x − 2x2 + 1; h(x) = x3 + x2 − x + 1;

k(x) = −x3 + 5x2 + x + 1 and j(x) = x3 − x; ∀x ∈ R:

a) Show that g and h belong to U.

b) Show that k ∈ span{g; h}.

c) Show that j =∈ span{g; h}.

d) Show that span{g; h} ̸= span{g; h; j}.

## Homework Equations

I don't really know any equation relevant, I do not really understand the concept behind Polynomial spans.

I am unsure about this:

is the span of a polynomial...lets say x^2+5x-3 and x^2+3x+10 is the span just span{x^2,x,-4, and x^2,x,10}?

I think you can see where I tried to attempt at this question (B), since both are x^3, x^2, x type of thing I said it was in the same span and etc.

My method (probably wrong) worked for (C), however I got stuck on (D), where

I am not sure what the span means.

## The Attempt at a Solution

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