Is this a subspace?

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.

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Originally Posted by splash

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?

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Let <a1,a2,a3,a4> be an element in this subset.
Does it mean that k<a1,a2,a3,a4>=<ka1,ka2,ka3,ka4> is in this subset?

Yes, because,
a1*a2*a3*a4 >= 0
Means that,
(ka1)*(ka2)*(ka3)*(ka4)=k^4(a1*a2*a3*a4)>= 0 because k^4>=0.

And if <b1,b2,b3,b4> is another element in subset.
Does it mean that <a1+b1,a2+b2,a3+b3,a4+b4> in the subset?
No, because,
Consider,
<1,2,3,4> and <-2,-1,3,4>

Yet, their sum,
<-1,1,6,8> is not an element because (-1)(1)(6)(8)<0.

Thus, this does not form a subspace.

Thank you, that was extremely helpful. So do I just have to guess by trial and error to see if I can find to vectors that are in the subset but not their sum?

Quote:

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Originally Posted by splash

Thank you, that was extremely helpful. So do I just have to guess by trial and error to see if I can find to vectors that are in the subset but not their sum?

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Yes, you can trial and error if you want to show it is not a subspace.

Otherwise, you need to formally prove that it is closed under addition.