a show W is a subspace of P3 b find a basis of W Solutiona 1

a) show W is a subspace of P3.

b) find a basis of W.

Solution

a)

1. p(x)=0 is in W

2. Let, p and q be in W

(p+q)(1)+(p+q)(2)=p(1)+p(2)+q(1)+q(2)=0+0=0

(p+q)\'(1)=p\'(1)+q\'(1)=0+0=0

Hence, p+q is in W

3. Let c be a scalar and p in W

(cp)(1)+(cp)(2)=cp(1)+cp(2)=c(p(1)+p(2))=0

(cp)\'(1)=cp\'(1)=c*0=0

Hence, cp in in W

Hence W is a subspace of P3

b)

Let, p(x)=a+bx+cx^2+dx^3

p(1)+p(2)=0=a+b+c+d+a+2b+4c+8d=2a+3b+5c+9d=0

Hence, a=-(3b+5c+9d)/2

p\'(1)=b+2c+3d=0

b=-2c-3d

a=-(3b+5c+9d)/2=-(-6c-9d+5c+9d)/2=c/2

p(x)=c/2+(-2c-3d)x+cx^2+dx^3=c(1-2x+x^2)+d(-3x+x^3)

Hence basis of W is

{1-2x+x^2,-3x+x^3}