To test whether $b\in\operatorname{span}\{v_1,\ldots,v_k\}$, row-reduce $[v_1\ \cdots\ v_k\ b]$. Membership holds iff there is no inconsistent row $[0\ \cdots\ 0\ c]$ with $c\ne0$. Equivalently, adjoining $b$ does not increase rank. A solution gives explicit combination coefficients.
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