Question

Is W a subspace of V?

Answer 1

\[V=\mathcal{P}_2\]

Answer 2

\[W=\left\{ a+bx+cx^2 : a+b+c=0 \right\}\]

Answer 3

I know it is a subspace if it closed under addition and scalar multiplication

Answer 4

it's the condition\[a+b+c=0\] that's making it difficult for me to show closure under addition

Answer 5

let say to arbitrary x1, x2 in W such that f(x1) = a1 +b1x +x1x^2
f(x2) = a2 +b2x +c2x^2
now x1+x2 and f(x1+x2) = (a1+a2 )+(b1+b2)x +(c1+c2)x^2 satisfy (a1+a2)+(b1+b2) +(c1+c2) =0 or not? trivially it is =0, so, W is close under addition

Answer 6

Thanks, I was very close; your reply made it clear!

Answer 7

np