Homework SourseDo the vectors x21 x2x1 x form a basis of P2 If yes explain
Do the vectors x^2+1, -x^2+x-1, x form a basis of P2? If yes, explain why. If not, give a basis for the set spanned by those vectors.
Do the vectors x^2+1, -x^2+x-1, x form a basis of P2? If yes, explain why. If not, give a basis for the set spanned by those vectors.
Solution
Let p(x) be an arbitrary polynomial in P2. Then p(x) = ax2 + bx + c, where a,b,c are arbitrary constants. Now, 2a(x2 +1) +a(- x2+x-1) + (b-a)x +(c-a)=2ax2 + 2a–ax2+ ax –a + bx –ax + c–a =(2ax2 - ax2 )+(ax + bx –ax) + (2a –a + c –a) = ax2 + bx + c = p(x). Thus,the vectors x2 +1, - x2+x-1, x form a basis of P2.
