Results 1 to 10 of 10

Math Help - Subset, Subspaces and Linear Algebra

  1. #1
    Junior Member
    Joined
    Dec 2008
    Posts
    51

    Subset, Subspaces and Linear Algebra

    i have a question about subspaces, is the following subset (lets call it W)
    a subspace of (let's call it V)

    W=\{A\in V |A^T = -A\}

    and V is a two by two matix

    hope someone understands
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by iLikeMaths View Post
    i have a question about subspaces, is the following subset (lets call it W)
    a subspace of (let's call it V)

    W=\{A\in V |A^T = -A\}

    and V is a two by two matix

    hope someone understands

    Question: If V is the vector space consisting of all two by two matrices, then is W a subspace??

    Useful Fact: W is a subspace of V iff for any scalar \alpha, \forall A, B \in W, \alpha A + B \in W

    Idea: Observe that (\alpha A + B)^T = \alpha A^T + B^T =-(\alpha A + B)

    So.....?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2008
    Posts
    51
    could you explain a bit more please, i am totally clueless
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by iLikeMaths View Post
    could you explain a bit more please, i am totally clueless
    Do you know that W is a subspace of V iff for any scalar \alpha, \forall A, B \in W, \alpha A + B \in W?

    Its easy to prove, just check all the axioms by an appropriate choice of alpha.

    The idea shows that \alpha A + B is of the form that belongs to W.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Dec 2008
    Posts
    51
    where did B come from i thought we were dealing with A^T and -A
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Dec 2008
    From
    Indiana
    Posts
    127
    Quote Originally Posted by iLikeMaths View Post
    i have a question about subspaces, is the following subset (lets call it W)
    a subspace of (let's call it V)

    W=\{A\in V |A^T = -A\}

    and V is a two by two matix

    hope someone understands
    Quote Originally Posted by Isomorphism View Post
    Do you know that W is a subspace of V iff for any scalar \alpha, \forall A, B \in W, \alpha A + B \in W?

    Its easy to prove, just check all the axioms by an appropriate choice of alpha.

    The idea shows that \alpha A + B is of the form that belongs to W.
    Quote Originally Posted by iLikeMaths View Post
    where did B come from i thought we were dealing with A^T and -A
    W=\{A\in V |A^T = -A\}<br />
    means any element, say A, in W has to satisfy A^T = -A. It does not mean that it is the only thing in the set. So if B is in W, then B^T = -B.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Dec 2008
    Posts
    51
    can any one provided a step by step explaination to the original question because i really need to understand this? do i find the matrix A, please help
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Dec 2007
    From
    Melbourne
    Posts
    428
    To prove that a subset W is a subspace you need to show:
    A\in W and B\in W \implies A+B \in W
    For any scalar \alpha, A\in W \implies \alpha A \in W
    W is non-empty.

    For the first statement, assume that A^T = -A and B^T = -B and show that (A+B)^T = -(A+B). Use a similar technique for the second statement. You could also do these two in a single step if you follow Isomorphism's method. For the last statement, you simply need to find a 2 by 2 matrix that satisfies A^T = -A.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Dec 2008
    Posts
    51
    the zero vector satisfies A^T = -A hence W is a subspace of V, is this correct
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Senior Member
    Joined
    Dec 2007
    From
    Melbourne
    Posts
    428
    the zero vector satisfies hence W is a subspace of V, is this correct
    It is certainly a start. But to conclude that W is a subspace of V you need to prove all 3 of the statements I listed before or prove Isomorphism's statement and do the bit you have already done.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help on some Linear Algebra - subspaces
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: February 8th 2011, 08:51 PM
  2. Subspaces--Linear Algebra
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 2nd 2009, 05:16 PM
  3. Linear Algebra Help...Subspaces?
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 8th 2008, 08:13 AM
  4. Linear Algebra Help...Subspaces?
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: September 7th 2008, 06:30 PM
  5. Linear Algebra: Subspaces
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: November 1st 2007, 09:10 PM

Search Tags


/mathhelpforum @mathhelpforum