What is the dimension of the subspace of $P_2$ given by
$ W = span \{2+ x^2, 4-2x+3x^2, 1+x\} ?$
I know dimension is the number of vectors in the basis, so I want to know if W's vectors are each linearly independent...
But I have no convenient way of telling if the vectors are linearly independent. The Wronskian is no good - it just evaluates to 0 for various substitutions ($x=0,1)$, which doesn't give me information. I'm looking for an efficient way to answer this problem.
I've also considered the rank-nullity theorem. dim$(P_2) = 3$, but does that tell me anything about the dim$(W)?$
Thanks in advance.