Results 1 to 5 of 5

Math Help - Which of the following subsets are subspaces?

  1. #1
    Newbie
    Joined
    Nov 2011
    Posts
    12
    Thanks
    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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762

    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.

    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2011
    Posts
    12
    Thanks
    1

    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762

    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2011
    Posts
    12
    Thanks
    1

    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?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subsets that are subspaces of P2
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 17th 2011, 09:02 PM
  2. Which subsets of R^3 are subspaces?
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 21st 2010, 01:28 AM
  3. Replies: 3
    Last Post: May 4th 2009, 12:35 AM
  4. Verifying Subsets as Subspaces through scalar vector
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: April 26th 2009, 12:38 AM
  5. Subsets and subspaces, perpendicular sets!
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 7th 2008, 12:51 AM

Search Tags


/mathhelpforum @mathhelpforum