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Let $V$ be a finite dimensional vector space over an infinite field $\mathbb{F}$, and let $v_1,\dots, v_n \in V$ be a collection of non-zero vectors for which $\dim \text{span}\{v_i, v_j\} =2$ for all $i\neq j \in \{1,\dots, n\}$ (i.e. they are in 2-general position) and $\dim \text{span} \{v_1,\dots, v_n\} \geq 3$. Does there always exist an index $i \in \{1,\dots, n\}$ and a linear map $\Pi \in L(V)$ with $\ker(\Pi)=\text{span}(v_i)$ such that $\dim \text{span} \{\Pi v_j, \Pi v_k\} =2$ for all $j \neq k \in \{1,\dots, n\} \setminus \{i\}$?

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No. The vectors \begin{align} \begin{bmatrix} 1\\0\\0 \end{bmatrix} , \begin{bmatrix} 0\\1\\0 \end{bmatrix} , \begin{bmatrix} \frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\\0 \end{bmatrix} , \begin{bmatrix} 0\\0\\1 \end{bmatrix} , \begin{bmatrix} 0\\\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}} \end{bmatrix} \end{align} in $\mathbb{R}^3$ do not satisfy this property.

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