Do you have the theorem that "any linearly independent set containing a number of vectors equal to the dimension of the space is a basis" and that the dimension of P2 is 2? If so, then yes, showing that those three vectors are independent show this is a basis. If not, then you should show that for any a, b, c, there exist , , such that . That's, fairly easy- just solve for , , and .
No, your "vector" is a+ b+ cx^2 for any a, b, c- you can't just make a= b= c= 1. Your coefficients must be in terms of a, b, and c- just what I said in my last sentence above! Find , , such that , for all x.(b) I know this is incorrect, but I got:
p(x) = a(1,0,0) + b(-1,1,0) + c(1,0,1)
a=b=c=1 (from (a))
One way to do this is to equate like coefficients. Multiplying out the left side, for all x so we must have , and . Putting those into the first equation, so :
Use the answer to (b) to write out and then . The "coordinates" of each are the number , , and(c) No idea?