Vector space of polynomials problem

1. Oct 29, 2007

captainjack2000

1. Consider the vector space of polynomials 1+x^3 , 1-x+x^2, 2x, 1+x^2
Are they linearly dependent or independent? dimension of vecotr space spanned by these vectors?

3. I have tried to solve this by letting
a1 = 1+x^3
a2 = 1-x+x^2
a3 = 2x
a4 = 1+x^2

Then I let
(alpha)a1 + (beta)a2 + (gamma)a3 + (delta)a4 = 0
(alpha)(1+x^3) + (beta)(1-x+x^2) + (gamma)(2x) +(delta)(1+x^2) = 0
(alpha +beta+delta) + x(2gamma - beta) + x^2(beta + delta) + x^3(alpha) = 0

So alpha + beta+ delta = 0
2gamma - beta = 0
beta + delta = 0
alpha = 0

But I can't solve for beta delta and gamma so how do I know if their independent or dependent?

2. Oct 29, 2007

Dick

There's no unique solution, so you'll have to write the solution in terms of some of your other variables. In this case it looks easy to solve for the other variables in terms of beta. Then ask yourself if there are any nonzero solutions.

3. Oct 29, 2007

captainjack2000

Vectors!

In that case beta = -delta
and gamma = beta/2
and alpha = 0

How can you tell if they are non-zero?
How do you find the dimension?

4. Oct 29, 2007

Dick

That gives you a solution for every value of beta. So putting beta=1 tells you what about linear independence?