vector space

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[shrewd inlet]

D

{u, v, w} is linear independent I'd assume.

Then clear it's not A. as if it were it'd mean that :
u = av + bw for some a, b in the field.
=> av + bw + (-1)u = 0
contradicting linear independence

C is a direct no-go from linear independence

B is false as if too as :
u + v + (-1)(u+v) = 0

now D is True as it just depends on the independence of {u, v, w}

you can even check :
suppose otherwise, hence au + b(u+v) + c(u+v+w) = 0
=> (a+b+c)u + (b+c)v + (c)w = 0
now we see that if that had been True for a, b, c not all zero then it'd imply that (a+b+c), (b+c), (c) were not all zero either! you can check this too as them being all zero would imply that, c = 0, b+c=0,a+b+c=0 => b=0 => a=0. So it being linearly dependant implies that {u, v, w} is too so it must not be the case