Coordinates of antisymmetric matrix

[Physicsissuef]

Homework Statement

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

$$E_1=\begin{bmatrix} 0 & 1 & 1\\ -1 & 0 & 0\\ -1 & 0 & 0 \end{bmatrix}$$

$$E_2=\begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 1\\ -1 & -1 & 0 \end{bmatrix}$$

$$E_3=\begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & -1\\ 0 & 1 & 0 \end{bmatrix}$$

Homework Equations

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

The Attempt at a Solution

The matrix is equal to:

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

The base is $$B={(x_2+x_3,-x_1,-x_1) ; (x_3,x_3,-x_1-x_2) ; (-x_2,x_1-x_3,x_2)}$$

What should I do now?

 

[HallsofIvy]

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.

 

[Physicsissuef]

I understand. Thanks for the help, I wasn't sure what the problem was asking for...
I found $$(a_1,a_2,a_3)=(1,-3,0)$$ and in my book get (2,-2,1). Probably is my mistake, I will check again.