You are correct that the dimention of a space tells you how many linearly independant vectors are required to span a space.

In your example of it says that the basis must have 3 vectors in it

One basis could be

these are linearly independant and span

but you could also have the basis

these are also linearly independant and span

but they both still have 3 vectors.

One way to solve the above problem is to express the polynomials in terms of the standard basis of

The basis is So for example the vector . Now the vectors may look more familiar!