Dropping a Redundant Vector from a Spanning Family

Let $K$ be a field, let $V$ be a vector space over $K$, and let $n$ be a natural number. Inequalities between natural numbers use the order relations on $\mathbb{N}$.

Let $v$ be a tuple in $V^{n+1}$ that spans $V$, let $j$ be a natural number with $j\le n+1$, and let $v^{(j)}$ be the tuple in $V^{n}$ obtained from $v$ by extracting the index $j$, as defined in Extraction of a Summand from a Finite Sum of Vectors.

If $v_{j}$ lies in the span of $v^{(j)}$, then $v^{(j)}$ spans $V$.