Math Help - Which of the following subsets are subspaces?

1. Which of the following subsets are subspaces?

Hey guys,

I'm having trouble with subsets and subspaces (proper way of showing whether it's a subspace or not) I usually go over the 3 statements that have to be true for it to be a subspace, which is : 1. There exists a 0 vector 2. U and V such that U + V are in the subspace 3. KU is also in S. I'd really appreciate if somebody would go over the following questions with a step by step explanation.

And Also:
1. is the vector space of all real-valued functions defined on the interval , and is the subset of consisting of those functions satisfying
2. , and is the subset of consisting of those functions satisfying the differential equation
3. , and is the subset of all upper triangular matrices.

Thank You!

2. Re: Which of the following subsets are subspaces?

you need to define what kind of function p(t) is, which vector space we're taking as our "parent" space, for the first 3 sets.

we check for the 3 conditions:

a) the 0-function (which is constant for all x in [a,b]) has the property that 0(a) = 0(b), since both of these are 0, so it is in S.
b) suppose f,g are in S. we want to show that f+g is in S. now, since f and g are in S, f(a) = f(b), g(a) = g(b). therefore:
(f+g)(a) = f(a) + g(a) = f(b) + g(b) = (f+g)(b), so f+g is in S.
c) let k be a real number, and left f be any element of S. then (kf)(a) = k(f(a)) = k(f(b)) = (kf)(b), so kf is in S.

so S is indeed a subspace of V.

3. Re: Which of the following subsets are subspaces?

I apologize P(t) is a 2nd degree polynomial subspace, and V is a vector space "S" is a subset of V.

4. Re: Which of the following subsets are subspaces?

the procedure is still the same:

if $S = \{p(t) \in P_2(\mathbb{R}) | \int_0^4 p(t)\ dt = 0\}$ then clearly, the 0-polynomial is in S, since the definite integral over any interval of 0 is 0.

suppose p,q are in S. is p+q in S? let us see....

$\int_0^4 (p+q)(t)\ dt = \int_0^4 p(t) + q(t)\ dt = \int_0^4 p(t)\ dt + \int_0^4 q(t)\ dt = 0 + 0 = 0$

since p and q are both in S.

what about kp? again, we check:

$\int_0^4 (kp)(t)\ dt = \int_0^4 k(p(t))\ dt = k\int_0^4 p(t)\ dt = k(0) = 0$

since the last integral is 0 (because p is in S). so kp is in S, whenever p is.

we have satisfied all 3 conditions for a subspace, so S is a subspace of V.

5. Re: Which of the following subsets are subspaces?

I'm having trouble with "0" Polynomial, does 0 polynomial mean P(t)=0?

For example S={ p(t) | p'(8)=p(5) }
1. There exists a 0 polynomial such that P'(8)=P(5) <-- Correct?
2. Let P and F in S (P+F)'(8)=P'(8)+F'(8)=P(5)+F(5)=(P+F)(5) <-- Correct
3. P'(k8)=kP'(8)=kP(5)=P(k5) for any K in IR

So the above is a subspace.

Could someone correct me if I'm wrong?