v= 1+x^{2}

q= x^{2}

r= -2+2x^{2}

s= -3x

dim P_{2}= 3 (c1, x, x^{2})

c1 c2 2c3 0 0

0 0 0 -3c4 0

c1 0 -2c3 0 0

I know a basis in linearly independent c1 = c2 = c3 = c4 = 0

I know that to span means the minimum amount of vectors required to "encompass" a subspace without a vector being a linear combination of the others, and therefore if one of the vectors is removed the vectors wouldn't account for all the possible combinations in the space.

I think I'm trying to show that I can remove one of these vectors and still account for all the combinations....or maybe I have no idea what I'm doing. Do I just start removing vectors and see if they are linearly independent and spanning? Just looking seems like I could removeqandr. Please help me get started with the tactics of this problem. Thanks

Anthony