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A problem with basis and dimension

  1. Nov 17, 2009 #1

    I am working on the following problem:

    If a matrix is antisymmetric (thus A^T = -A), show that

    P = {A [tex]\in[/tex] R | A is antisymmetric} is a subset of Rnxn. Also, find the dimension of P.

    So far, I've proven that P is a subset of R and I am guessing that, in order for me to find the dimension, I need to find the basis of P first. Here is where I am kind of stuck.

    I understand what the properties of the base would be (1. the vectors inside the set will be linearly independent and 2. the basis will be a spanning set for P), but how exactly should I start working so that I can find the basis itself...

    A hint would be highly appreciated!

    Thanks a bunch guys!
  2. jcsd
  3. Nov 17, 2009 #2
    How many numbers do you need to fully describe an arbitrary antisymmetric nxn matrix?

    In other words, how many independent components does such matrix have?
  4. Nov 17, 2009 #3
    Well, I am debating between n and 2n numbers.
    The following 2x2 matrix is antisymmetric:
    0 a
    -a 0

    So, it depends on one number, a.

    A description of a 3x3 matrix would depend on 3 numbers, a, b, and c and so forth.

    Does this mean that this is the answer to the problem? n elements?

    Thank you in advance!
  5. Nov 17, 2009 #4
    Try to write down an asymmetric 4x4 matrix.
  6. Nov 17, 2009 #5
    Thanks for pointing that out.

    For 2x2 matrices,1 element is necessary.
    For 3x3 matrices, 3 elements, for 4x4 - 6 elements, for 5x5 - 9 elements, for 6x6 it's 15 elements..

    I am trying to come up with the proper function that will generate such output, given the input (n), but ... I can't really think of anything..
  7. Nov 17, 2009 #6

    Consider that, say, a 6x6 matrix has 6 elements on the diagonal, 15 elements above the diagonal, and 15 elements below the diagonal. 6+15+15 = 36.
  8. Nov 17, 2009 #7
    Oh great!
    So can we say then that the dimension for P would be:
    (n^2 - n)/2?

    Thank you so much!!!!
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