Thread: 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!

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.

i will address number 1:

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.

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

the procedure is still the same:

if 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....



since p and q are both in S.

what about kp? again, we check:



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.

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?