Let W p P3 p1 p2 0 p1 0 a Show W is a subspace of P3 b

Let W = {p P₃ : p(1) + p(2) = 0, p\'(1) = 0}.

(a) Show W is a subspace of P₃.

(b) Find a basis of W.

Solution

a)

1. 0 belongs to this set

2. Let, p,q belong to this set

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

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

Hence it is a subspace

b)

General polynomial in P3

is :f(x)= a+bx+cx^2+dx^3

f\'(x)=b+2cx+3dx^2

f\'(1)=b+2c+3d=0 hence, b=-2c-3d

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

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

f(x)= a+bx+cx^2+dx^3=3c/2-(2c+3d)x+cx^2+dx^3=(3/2-2x+x^2)c+(-3x+x^3)d

So basis is {3/2-2x+x^2,-3x+x^3}