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Let $\mathcal{R}$ be a ring and let $v^0,\ldots,v^{k-1}\in\mathcal{R}^m$ with $m \geq k$. Suppose we wish to find $w\in Span(v^0,\ldots,v^{k-1})$ such that $k-1$ specified coordinates of $w$ vanish (say $w[j_0] = \ldots = w_[j_{k-2}] = 0$). Then the following determinantal construction for $w$ does the trick:

$w = \sum_{i=1}^k (-1)^k\cdot\det\begin{pmatrix}{v^0}[j_0]&{v^0}[j_1]&\ldots&{v^0}[j_{k-2}]\\\\{v^1}[j_0]&{v^1}[j_1]&\ldots&{v^1}[j_{k-2}]\\\\\vdots&\vdots&\ddots&\vdots\\\\\widehat{v^i[j_0]}&\widehat{v^i[j_1]}&\ldots&\widehat{v^i[j_{k-2}]}\\\\\vdots&\vdots&\ddots&\vdots\\\\{v^k}[j_0]&{v^k}[j_1]&\ldots&{v^k}[j_{k-2}]\end{pmatrix}\cdot v^i$

The individual coordinates of $w$ also have a nice determinantal form (which follows from the above expression):

$w[i] = \det\begin{pmatrix}v^0[i]&v^0[j_0]&v^0[j_1]&\ldots&v^0[j_{k-2}]\\\\v^1[i]&v^1[j_0]&v^1[j_1]&\ldots&v^1[j_{k-2}]\\\\\vdots&\vdots&\vdots&\ddots&\vdots\\\\v^{k-1}[i]&v^{k-1}[j_0]&v^{k-1}[j_1]&\ldots&v^{k-1}[j_{k-2}]\end{pmatrix}$

For example, suppose we start with the 3 vectors $v^0 = \langle 2,3,5,7,11,13\rangle$, $v^1 = \langle17,19,23,29,31,37\rangle$ and $v^2 = \langle 41,43,47,53,59,61\rangle$ and wish to find a vector $w$ in their span whose final 2 coordinates are 0. Then the above formulas give us the following vector:

\begin{align*} w &= \det\begin{pmatrix}31&37\\\\59&61\end{pmatrix}\cdot v^0 - \det\begin{pmatrix}11&13\\\\59&61\end{pmatrix}\cdot v^1 + \det\begin{pmatrix}11&13\\\\31&37\end{pmatrix}\cdot v^2\\\\ &= -292\cdot v^0 + 96\cdot v^1 + 4\cdot v^2\\\\ &= \left\langle\det\begin{pmatrix}2&11&13\\\\17&31&37\\\\41&59&61\end{pmatrix},\det\begin{pmatrix}3&11&13\\\\19&31&37\\\\43&59&61\end{pmatrix},\det\begin{pmatrix}5&11&13\\\\23&31&37\\\\47&59&61\end{pmatrix},\right.\\\\ &\;\;\;\;\;\;\;\;\;\;\left.\det\begin{pmatrix}7&11&13\\\\29&31&37\\\\53&59&61\end{pmatrix},\det\begin{pmatrix}11&11&13\\\\31&31&37\\\\59&59&61\end{pmatrix},\det\begin{pmatrix}13&11&13\\\\37&31&37\\\\61&59&61\end{pmatrix}\right\rangle\\\\ &= \langle 1212, 1120, 936, 952, 0, 0\rangle \end{align*}

These formulas are easily derived/proved using Cramer's rule and/or other methods involving exterior products (which is how I came up with them when I was trying to construct such a vector), and like Cramer's Rule are rather beautiful, so I would be surprised if they are not already written down/used somewhere. Nevertheless, I don't recall having ever seen such constructions in any Linear Algebra books.

Main Question: Has anyone seen either of the above equivalent formulas before? If so, do they have a name?

Secondary Question: Whether or not they have a name, has anyone seen these formulas used as part of any other proofs?

Update: After thinking more about Question 2 over the last few days I realized this sort of construction $could$ arise in some group representation problems where one wants to explicitly find a character which is zero on certain conjugacy classes, such as when one is calculating induced representations. But I have definitely never seen it in any such character-theoretic arguments, and it is not even immediately clear to me whether it would necessarily give non-virtual representations in general. On the other hand, studying for a given group $G$ which sets of characters paired with which sets of conjugacy classes give rise to pure representations via this construction (aside from the trivial cases of pairs obtained from induction of representations) sounds intriguing enough that I will probably take a look at it. Still not clear if the construction has actually been used in any such proofs elsewhere though...

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I have seen it ... In exercises I gave Linear Algebra students about cramer's rule :-)

As to the representation theory connection - you certainly can't expect to get non virtual representations in general, since in many cases such a character simply doesn't exist (e.g. You want it to vanish on all regular semi simple elements in GL2 over finite field).

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