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Let there be given set of vectors $V = \{v_1, v_2, ..., v_n\}$ and set of vectors $S = \{s_1, s_2, ..., s_k\}$ where $n > k$. The set of vectors $V$ can be constructed by $S$ if the vectors in $V$ lie in the linear hull/span of $S$. In order for this to be possible, the vectors in $V$ must not be linearly independent (because $n > k$). In fact, there may only be as many linearly independent vectors in $V$ as there are in $S$. However, this condition is necessary but not sufficient as the vectors in $V$ and $S$ can span different subspaces.

Is there some algorithm to find out whether $V$ can be constructed by $S$? (is there some implementation also?)

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    $\begingroup$ Do you know how to compute the rank of a matrix? Have you heard of row echelon form? $\endgroup$
    – John L.
    Commented Apr 9, 2022 at 0:14
  • $\begingroup$ Sure, thanks. But is there some way that is more robust, that will say to what extend (that will say whether at least approximately) $V$ can be constructed by $S$ ? $\endgroup$
    – Eauriel
    Commented Apr 9, 2022 at 13:02

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