# Coordinates of antisymmetric matrix

## 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?

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.