1) Let S be the set of vactors <a1, a2, a3, a4> in R^4 such that a1a2a3a4 is greater than or equal to 0. Is S a subspace of R^4?

2) Let S be the set of vectors <b1, b2, b3> in C^3 such that 2(b1)-(3-2i)(b2)+7i(b3)=0 and (1-i)(b1)-(5+4i)(b3)=0. Is S a subspace of C^3?

3) Let S be the set of polynomials P with real coefficients and with the property that P(1)=2 Is S a subspace of P(R) --the set of all polynomials wit hreal coefficients

4) Let S be the set of three times differentiable functions f:R->R satisfying the differential euqation y'''-2y''+3y'-4y=0. Is S a subspace of F(R) --the set of all functions f:R->R

5) Let S be the set of vectiors <a+2b-3c+5d, 7a-6c+12d, -4a+b+d, 9c-11d> for all a,b,c,d in C. Is S a subspace of C^4?

It would be great if I could get some detailed help with some of these specific questions, but I guess my general problem is that I don't know how to determine if a given set of vectors is a subspace. Are there steps or things I can look for every time? Any help is appreciated. Thanks.