Let $B=(b_1,\ldots,b_n)$ and $c=\sum_i\gamma_ib_i$. Replacing $b_j$ by $c$ yields a basis iff $\gamma_j\ne0$. If it is nonzero, solve for $b_j$ using $c$ and the remaining basis vectors; if zero, $c$ lies in the span of the remaining vectors and the new list is dependent. This is a basis-exchange criterion.
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