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Basis question

  1. Mar 16, 2007 #1
    This is the problem that im working on.

    Find a basis for the vector space of all 3x3 symetric matricies.

    Is this a good place to start

    111
    110
    100

    using that upper triangular

    then spliting it into the set.

    100 010 001 000 000 000
    000 000 000 100 010 000
    000 000 000 000 000 100

    would that work?

    i know that it has to be linearly independent and span.
     
    Last edited: Mar 16, 2007
  2. jcsd
  3. Mar 16, 2007 #2

    matt grime

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    Are any of the matrices you wrote out symmetric?

    The matrix you started from isn't upper triangular, by the way. Why would you want to start with that matrix anyway? A symmetric matrix is one satisfying A=A^t, so take the 3x3 matrix

    [a,b,c]
    [d,e,f]
    [g,h,j]

    and transpose it, equate the two, what conditions do you now have on a,b,..,j?
     
  4. Mar 16, 2007 #3
    I choose

    111
    110
    100

    because it was equal to its transpose.
    I know that b is equal to to d and so on for the transpose you mention.

    I choose the six randomly as a guess.

    seeing what happens to the above matrix you gave me though, how about this?

    100
    000
    000 becuase a doesnt move

    010
    100
    000 those have to be there

    001
    000
    100 same here

    000
    000
    001 and this

    000
    001
    010

    I guess that im just looking for a pattern. Those seem better, I feel like there should be six though.
     
  5. Mar 16, 2007 #4

    radou

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    One matrix is missing.
     
  6. Mar 16, 2007 #5
    000
    010
    000

    right?
     
  7. Mar 16, 2007 #6

    radou

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    Yes, right.
     
  8. Mar 17, 2007 #7
    Ok. So here is where my new issue arrises.

    For this set of matricies to be a BASIS for all symertric 3x3 matricies then it has to linearly independent... right?

    So I took my set.

    100 010 000 001 000 000
    000 100 010 000 001 000
    000 000 000 100 010 001

    and set it equal to a 3x3 zero matrice, and set up a system with scalars.
    They end up all equaling zero... the trivial solution. Nice.

    But they have to check out for span also to be considered a basis right?

    That means that each one has to be a linear combination of the other five remaining right?

    Well for some of them I get 0 = 1 for a solution

    Anyone know where im going wrong?

    Thanks, Rob
     
  9. Mar 17, 2007 #8

    matt grime

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    Yes. It is this line:

    Reread the definition of span. Cos you've just written something that implies linear dependence. It spans if every symmetric matrix can be written as a combination of them, and that is obviously true.
     
  10. Apr 3, 2007 #9
    would any one help me solve this problem?

    I read all posts above but still confuse.

    Much appreciate.
     
  11. Apr 4, 2007 #10

    matt grime

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    What is consfusing you? What is the generic symmetric matrix? Therefore what is a basis?
     
  12. Apr 4, 2007 #11
    I know 3x3 symmetric matrix formed by
    a b c
    d e f
    g h i
    and b=d c=g f=h
    but I don't know how to start on this case.
    What I'd write at beginning on my homework paper?
     
  13. Apr 4, 2007 #12

    matt grime

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    Write that as a sum of matrices with lots of zeroes.
     
  14. Apr 4, 2007 #13
    That array just refuses to cooperate...
     
  15. Apr 6, 2007 #14

    daniel_i_l

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    To find a basis you just do three steps:
    1) find the generic matrix (or vector in the general case)
    2) break it up into a number of matricies where each one containes only one type of parameter with everything else zeros(but no two contain the same type)
    3) turn the parameters in each matrix into ones.
     
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