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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) = 2x^3− x − 2x^2+ 1;

h(x) = x^3+ x^2− x + 1;

k(x) = −x^3+ 5x^2+ x + 1 and

j(x) = x^3− 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

## The Attempt at a Solution

My Attempt:

I have no idea how to do a) I tried plugging -1 and +1 in g and h however it doesn't meet the requirement of f(1)=f(-1).

I am sure I know how to do party b,c and d. However, I am doing something wrong and have no clue what it is...

So what I did for b) was:

k = a(g) + b(h)

after that you get something like (....)x^3 + (....)x^2 + (....)x + (a+b)

So I set the coefficient (....) equal to the coefficients of k (-1,5,1,1)

however I am unable to solve the linear system.

But it says it is in the span so I don't know what happened maybe mtah error, but I redid it a lot of times.