Let W = {(f(x)∈ P2[R] : f '(x) + xf(0) = 0}
i) Prove that W is a subspace of P2[R].
ii) Find a basis for W.
Here's what I have so far:
i) I have to verify that the subspace (1) contains the zero space, (2) is closed under addition, and (3) is closed under scalar multiplication.
i)
(1) Contains the zero space
f'(x) + xf(0) = 0
f'(0) + (0)f(0) = 0
0 = 0,
so 0 ∈ W
I'm stuck on numbers 2 and 3, because I'm not sure how to represent f '(x) + xf(0) = 0 with examples that I can just plug in and show that it is closed under addition and scalar multiplication.
I generally do these problems with vectors or the like - for example, if the question was
Let Let W = {(x, y, z)∈ R^3 : x + y - 2z = 0}
and I wanted to show if it was a subspace, I would just show from some vectors v1 = (a1, a2, a3) and v2 = (b1, b2, b3) that
(a1 + b1) + (a2 + b2) - 2(a3 + b3) = (a1 + a2 - 2a3) + (b1 + b2 - 2b3), thus is closed under addition
and
c(a1 + a2 - 2a3) = ca1 + ca2 - 3ca3, thus is closed under scalar multiplication.
and I would find the basis by just finding the kernel of x + y - 2z = 0
I'm not sure how to deal with the problem when it is converted into polynomials. Thanks.