Results 1 to 4 of 4

Math Help - [SOLVED] Show that the following subsets U and W are subspaces of V

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    90

    [SOLVED] Show that the following subsets U and W are subspaces of V

    Let V be a vector space of nxn Matrices over a field K. Show that the following subsets U and W are subspaces of V.

    1. U = {A = (a_ij) \in V | a_ij = a_ji, \forall 1 \leq i, j \leq n}.

    2. Let T be a fixed matrix in V and W = {A \in V | AT = TA}.

    So, A are symmetrical matrices in \mathbb{R}^n, is there anything special about symmetrical matrices relevant for this question (part1)?

    If the matrix is symmetrical than (part2) AT \equiv TA?
    Last edited by bmp05; May 3rd 2009 at 01:57 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517
    A subspace is just a vector space that is contained. You just gotta show that for both they are closed under addition and scalar multiplication. Also, that they contain the additive identity, but this is trivial for both of them.

    1) This is simply the set of symmetric matrices. A^T=A
    Let A and B be symmetric matrices.
    Addition:
    a_{ij}=a_{ji} and b_{ij}=b_{ji} by definition.
    A+B you just add compnantwise, so the ijth entry of A + B is (A+B)_{ij}=a_{ij} + b_{ij} = a_{ji} + b_{ji} = (A+B)_{ji} (check)
    Scalar multiplication.
    A is a symmetric matrix so a_{ij}=a_{ji}. Let k be a scalar from the field K.

    ka_{ij}=ka_{ji} so kA is a symmetric matrix.
    (check)
    Clearly the 0 matrix is symmetric every entry is 0.
    (check)
    1) is a subspace.



    2)T is fixed and this is just the orbit of T under conjugation (This first statement is only true in the set of invertible matrices, but I never used invertibility in the proof, I meant to remove this statement actually, but forgot). You are looking at the set of all matrices A such that AT=TA call it W.

    Let A and B be matrices in W. Then AT=TA and BT=TB
    Addition
    (A+B)T=AT +BT= TA + TB=T(A+B)
    so A+B is in W
    (Check)
    Let k be an element in the field.
    T(kA)=(Tk)A=k(TA)=k(AT)=(kA)T
    so kA is in W.
    (check)
    T(0)=0=(0)T
    so 0 is in W
    It is a subspace.

    Done and done.
    Last edited by Gamma; May 3rd 2009 at 08:29 PM. Reason: Conjugation clarification.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2009
    Posts
    90
    Thanks for the reply, Gamma. So, I just use the vector space rules to show that they are valid for these subspaces. I thought, maybe there is something special about this type of matrix. Thanks again!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517
    Yup, that is all you have to do. You will find this to be the case a lot if you continue on in math. You will have to find things are subgroups of groups, subrings of rings, subfields of fields, subspaces of vector spaces, submanifolds of manifolds, it is a fairly common theme in mathematics. All these cases are examples of when you just need to make sure they are both a subset and satisfy the same rules as the mother object.

    But in a way, you should not dismiss the uniqueness of these particular types of matrices. Their structure is important for them to actually be subspaces. Not just any subset of matrices will give you a subspace.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Which of the following subsets are subspaces?
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: November 12th 2011, 03:47 AM
  2. Subsets that are subspaces of P2
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 17th 2011, 08:02 PM
  3. Which subsets of R^3 are subspaces?
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 21st 2010, 12:28 AM
  4. Verifying Subsets as Subspaces through scalar vector
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: April 25th 2009, 11:38 PM
  5. Subsets and subspaces, perpendicular sets!
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 6th 2008, 11:51 PM

Search Tags


/mathhelpforum @mathhelpforum