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Given a lower-triangular matrix $L \in \mathbf{R}^{p \times p}$ and vector $v \in \mathbf{R}^p$, how can I construct a basis $B$ for the subspace $S = \mathbf{R}^p / \operatorname{span}(\{v\})$ such that the projection of $L$ onto $S$ is lower-triangular in $\mathbf{R}^{(p-1) \times (p-1)}$ with respect to basis $B$?

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