Let P2 be the set of all polynomials of degree at most 2 wit

Let P_2 be the set of all polynomials of degree at most 2 with real coefficients, i.e., P_2 = {a_0 + a_1x + a_2x^2; a_0, a_1, a_2 R}. Show that {1 + x, x + x^2,1 + x^2} is a basis for P_2.

Solution

Show that the polynomials in the set B = { 1 +x , x+x^2 , 1+x^2 } are linearly independent.

You show that B spans P2 by showing how to write an arbitrary p(x) P2 as a linear combination of members of B, so let p(x)=a0 + a1x + a2x^2 P2 . You want to find coefficients c0, c1, c2 such that

p(x) = c1(1+x) +c1(x +x^2) +c2( 1 +x^2)

we show that B is a linearly independent set, try to do what the definition of linear independence says you must do to show that a set is linearly independent: assume that c0,c1 and c2 are scalars such that:

c0(1+x) +c1(x +x^2) +c2( 1 +x^2) =0

equations we get from equating coefficients from above : c1 +c2 =0

c0 +c2 =0

c0 +c1 =0

c1 +c2 =0

which implies c0 = c1 = c2 =0. the system has the unique solution.

This proves that P2 is a basis for P2