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Coordinates of antisymmetric matrix

  • #1

Homework Statement



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]


Homework Equations



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

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?
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,792
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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.
 
  • #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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