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Coordinates of antisymmetric matrix

  1. May 28, 2008 #1
    1. The problem statement, all variables and given/known data

    Let's say that V is the vector space of all antisymmetric 3x3 matrices. Find the coordinates of the matrix [tex]A=\begin{bmatrix}
    0 & 1 & -2\\
    -1 & 0 & -3\\
    2 & 3 & 0
    \end{bmatrix}[/tex] in ratio with the base:

    [tex]E_1=\begin{bmatrix}
    0 & 1 & 1\\
    -1 & 0 & 0\\
    -1 & 0 & 0
    \end{bmatrix}[/tex]

    [tex]E_2=\begin{bmatrix}
    0 & 0 & 1\\
    0 & 0 & 1\\
    -1 & -1 & 0
    \end{bmatrix}[/tex]

    [tex]E_3=\begin{bmatrix}
    0 & -1 & 0\\
    1 & 0 & -1\\
    0 & 1 & 0
    \end{bmatrix}[/tex]


    2. Relevant equations

    antisymetric matrix is only if [itex]A^t=-A[/tex]

    3. The attempt at a solution

    The matrix is equal to:

    [tex]f: \mathbb{R}^3 \rightarrow \mathbb{R}^3 , f(x_1,x_2,x_3)=(x_2-2x_3,-x_1-3x_3,2x_1+3x_2)[/tex]

    The base is [tex]B={(x_2+x_3,-x_1,-x_1) ; (x_3,x_3,-x_1-x_2) ; (-x_2,x_1-x_3,x_2)}[/tex]

    What should I do now?
     
  2. jcsd
  3. May 28, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Do you understand what the problem is asking? It's exactly like many other problems you have done in the past: write the given "vector" (the matrix A) as a linear combination of the given basis "vectors" (the matrices E1, E2, E3). That is, find numbers a1, a2, a3 so that A= a1E1+ a2E2+ a3E3. Those numbers are the "coordinates".

    Setting corresponding components on both sides equal will give you 9 equations, of course, but they should reduce to 3 independent equations for a1, a2, and a3. For example, the upper left component (A11) of every matrix is 0 so that just becomes 0= 0a1+ 0a2+ 0a3 which is satisfied by all numbers.
     
  4. May 28, 2008 #3
    I understand. Thanks for the help, I wasn't sure what the problem was asking for...
    I found [tex](a_1,a_2,a_3)=(1,-3,0)[/tex] and in my book get (2,-2,1). Probably is my mistake, I will check again.
     
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