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I am in a problem seminar class and I have not taken Linear Algebra in over 4 years so I am having a lot of problems with this. Please help...

Let

Let

Find a basis for

Not sure

I know that to be a basis, that the set must be linearly independent and span

I proved in the first part of this problem that V is a subspace of

I also said that in order for f(-2) = f(1),

a(sub0) - 2a(sub1) +4a(sub2) = a(sub0) + a(sub1) + a(sub2)

-3a(sub1) + 3a(sub2) so

a(sub1) = a(sub2)

not sure if this is right or not...

so I randomly chose 2 different elements of this set. and showed that they were linearly independent.

the elements I chose were v(sub 1) (x)= 1 + 3x +3x^2 and v(sub 2 ) (x) = 0 +1x + 1x^2

Now I found a theorem in my linear algebra book that says;

"Let H be a subspace of a finite-dimentional vector space V. Any linearly independent set in H can be expanded, if necessary to a basis for H. Also H is finit dimensional and dim H < or = dim V "

Now I know that dim

Thus dim

Am I going about this wrong? I don't really understand how to find span

Please point me in the right direction!

## Homework Statement

Let

**P**be the set of all polynomials with real coefficients and of degree less than 3. Thus,**P**= {f:f(x)= a(sub0) +a(sub1)x +a(sub2)x^2, a(sub i ) is in the reals}**P**is a vector spaces over the field of Reals under the usual opperations of addition and scalar multiplication of polynomials.Let

**V**= {f element of**P**: f(-2) =f(1)}Find a basis for

**V**.## Homework Equations

Not sure

I know that to be a basis, that the set must be linearly independent and span

**V**.## The Attempt at a Solution

I proved in the first part of this problem that V is a subspace of

**P**I also said that in order for f(-2) = f(1),

a(sub0) - 2a(sub1) +4a(sub2) = a(sub0) + a(sub1) + a(sub2)

-3a(sub1) + 3a(sub2) so

a(sub1) = a(sub2)

not sure if this is right or not...

so I randomly chose 2 different elements of this set. and showed that they were linearly independent.

the elements I chose were v(sub 1) (x)= 1 + 3x +3x^2 and v(sub 2 ) (x) = 0 +1x + 1x^2

Now I found a theorem in my linear algebra book that says;

"Let H be a subspace of a finite-dimentional vector space V. Any linearly independent set in H can be expanded, if necessary to a basis for H. Also H is finit dimensional and dim H < or = dim V "

Now I know that dim

**P**= 3Thus dim

**V**< or = 3. and since the set of two numbers I used above were linearly independent, they should be able to be expanded to form the basis, but my text does not tell me how to do this...Am I going about this wrong? I don't really understand how to find span

**V**either.Please point me in the right direction!

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