Vector Space help needed

I've attached the relevant question. I'm not really sure what the steps I need to take are. Do the standard basis vectors work?

The standard basis does not work: (1,0,0) doesn't satisfy the equation, sadly.

Could you have a guess at the dimension (with reasoning if you can) before I attempt to help? Don't worry about the basis for now.

The standard basis does not work: (1,0,0) doesn't satisfy the equation, sadly.

Could you have a guess at the dimension (with reasoning if you can) before I attempt to help? Don't worry about the basis for now.

Would (0,1,1), (1,0,0.5) and (0,-1,-1) form a basis because they satisfy the equation?

(0, 1, 1) + (0, -1, -1) = 0, so these two vectors are not linearly independent.

The fact is, the vector space you are trying to find a basis for does NOT have dimension 3, so you aren't going to be able to find a basis with 3 elements.

It might help to look at it like this:

If you have a vector (a, b, c) in the vector space, then we know that

a+2b-2c = 0.

So suppose you are given a, b. What must c equal? (*)

So we know that (a, b, c) = (a, b, Aa+Bb) where (A, B) are constants you should have found in (*).

Rewrite it as $a {\bf A} + b {\bf B}$ where A, B are constant vectors.

At this point, what can we say about A, B?

Do they span our vector space? (If so, why, if no, why)?
Are they linearly independent?

So...

No. You haven't correctly written (a, b, c) in the form aA + bB

Still wrong. You need to write it in the for aA + bB where A, B are constant vectors. (i.e. don't involve a or b).

E.g. if you had (a, b, c) = (a, b, 7a + 9b), then you would need to rewrite this as a (1, 0, 7) + b (0, 1, 9).

Yes.