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## 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?