Let V P2R Let W denote the set of polynomials in V whose co

Let V = P2(R). Let W denote the set of polynomials in V whose coefficients sum to zero.

(a): Show that W is a subspace of V .

(b): Find a set of vectors that span W.

Solution

a)

1.

0 belongs to W

2.

Let, p =ax^2+bx+c ,q=rx^2+sx+t be in W

p+q=(a+r)x^2+(b+s)x+(c+t)

a+r+b+s+c+t=0

So, p+q is also in W

Hence, W is a subspace of V

b.

General polynomial in W is

p(x)=ax^2+bx+c

a+b+c=0

c=-a-b

p(x)=ax^2+bx-a-b=a(x^2-1)+b(x-1)

So vectors than span W are

{x^2-1,x-1}