Question

Why is the set {1, x, x^2} a basis for P_2 when a+bx+cx^2 = a(1)+b(x)+c(x^2)?
I understand that the this set spans P_2, but how do we know it's linearly independent?

Answer 1

Because no linear combinations of any two of those vectors will yield any of the others.

Answer 2

So, 1, x, and x^2 are all vectors, and k(1)+k(x)=x^2 is not consistent? Same thing with combinations of vectors in the set? I have trouble getting over the fact the something is a vector when it's not expressed in the form (a,b,c) or whatever.

Answer 3

other vectors in the set*

Answer 4

The graph of k(1)+k(x) will never equal the graph of x^2. Therefore they are linearly independent.