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If I have any skew symmetric matrix $\widehat{v}$ with elements

\begin{bmatrix} 0 & -v_{3} & v_{2} \\ v_{3} & 0 & -v_{1} \\ -v_{2} & v_{1} & 0 \end{bmatrix}

where $v\in \mathbb{R} ^{3}$ then what can we say about the rows of this matrix? Are they linearly independent? The span of these 3 row vectors( Not columns) will be a plane?

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Assuming $v\neq 0$, $0$ as an eigenvalue of the matrix has multiplicity $1$ so its eigenspace (which is the same as the nullspace of the matrix) has dimension $1$. Therefore, the matrix has (row) rank $2$ and the rows will span a plane.

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  • $\begingroup$ Can't even upvote your answer. -__- $\endgroup$ – mathuser001 Jun 6 at 18:04

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