Let $W = \{ A \in \mathbb{M}_3(\mathbb{R}) |A^T = -A \ \text{and }\sum_{j=1}^{3} a_{1j} =0\}$.
write down the basis for $W$
My attempt :
acoording to the given condtion $W = \begin{bmatrix} 1 &-1&0\\1&0&-1 \\ 0& 1 & 0 \end{bmatrix}$
Now $\begin{bmatrix} 1 &-1&0\\1&0&-1 \\ 0& 1 & 0 \end{bmatrix} = a\begin{bmatrix} 1 &0&0\\0&0&0 \\ 0& 0 & 0 \end{bmatrix} + (-b) \begin{bmatrix} 0 &1&0\\0&0&0 \\ 0& 0 & 0 \end{bmatrix} + d\begin{bmatrix} 0 &0&0\\1&0&0 \\ 0& 0 & 0 \end{bmatrix} + h \begin{bmatrix} 0 &0&0\\0&0&0 \\ 0& 1 & 0 \end{bmatrix} + (-f)\begin{bmatrix} 0 &0&0\\0&0&1 \\ 0& 0 & 0 \end{bmatrix}$
$\begin{bmatrix} a &-b&c\\d&e&-f \\ g& h & i \end{bmatrix}= aB_1 + (-b) B_2 + dB_3 +hB_4 +(-f)B_5$
Now here i got $5$ linearly independent elements in $W$.
Is my answer is coorect or not ??
Pliz verified and tell me
No, it is not correct. In what you wrote, $W$ stands for a set of matrices and also for an individual matrix. That doesn't make sense. Besides, your matrix $W$ does not belong to the set $W$.
The matrices of $W$ are those of the form$$\begin{pmatrix}0&a&-a\\-a&0&b\\a&-b&0\end{pmatrix},$$which is equal to$$a\begin{pmatrix}0&1&-1\\-1&0&0\\1&0&0\end{pmatrix}+b\begin{pmatrix}0&0&0\\0&0&1\\0&-1&0\end{pmatrix}.$$Therefore, a basis of $W$ will be$$\left(\begin{pmatrix}0&1&-1\\-1&0&0\\1&0&0\end{pmatrix},\begin{pmatrix}0&0&0\\0&0&1\\0&-1&0\end{pmatrix}\right).$$