Results 1 to 2 of 2

Math Help - Find the basis and it's dimension

  1. #1
    Junior Member
    Joined
    May 2009
    Posts
    71

    Find the basis and it's dimension

    Find a basis for

    W = {|a b|
    _____|c d|
    :a+d=b+c} and state the dimension of W.

    Showing my work:
    I thought one way to solve this would be to exhange values in the matrix:


    Then establish our vector set. Determine if the set spans for W. We then show linear independence...and therefore our basis.

    Problem is:
    I set up matrices like this:

    a |1 0| +b|0 0| c|0 0| +d|0 1|
    ...|1 0|....|0 0|...|0 0|....|0 1|


    I then took a+b=c+d
    and said: a-c=-d-b

    Anyway, some years ago I did some similar math and thought of this:

    Wy not replace
    |a b|
    |c d|

    with:
    |a a|
    |e e|

    I used e is epsilon. Please feel free to trash my "thought" process and do not follow my example to solve it as it need not be solved this way, that is: I am just trying what I remember from previous years.

    You could say I am kind of grapsing at straws here guys (and gals) and need some help!

    Please advise,
    Thanks.
    |
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,977
    Thanks
    1121
    Okay, here's the "trashing": Your "all 0" matrices make no sense. The "0 matrix" can't be a basis vector since any set of matrices containing the 0 matrix can't be independent.

    a+ d= b+ c gives one condition on the four numbers. You can solve for one of the numbers in terms of the other three. For example, you can write d= b+ c- a. I have no idea what you mean by "e= epsilon". What is epsilon?

    Here's how I would do the problem:

    Taking a= 1, b= c= 0, d= -1 so \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix} is in the space.

    Taking b= 1, a= c= 0, d= 1 so \begin{bmatrix}0 & 1 \\ 0 & 1\end{bmatrix} is in the space.

    Taking c= 1, a= b= 0, d= 1 so \begin{bmatrix}0 & 0 \\ 1 & 1\end{bmatrix} is in the space.

    The space has dimension 3 and those three matrices form a basis.

    Of course, you could also solve for b, c, or d and get a different, but equally correct, basis.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basis and dimension
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 4th 2010, 12:02 AM
  2. Basis and dimension
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 26th 2010, 07:43 AM
  3. What is the dimension of V ? Find a basis for V
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 13th 2010, 06:14 PM
  4. Find a basis and the dimension of the following subspaces:
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 23rd 2008, 05:35 PM
  5. Basis and Dimension
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 22nd 2007, 10:03 PM

Search Tags


/mathhelpforum @mathhelpforum