Spanning List of the Right Length is a Basis

Spanning list of the right length is a basis

Suppose $V$ is finite-dimensional. Then every spanning list of vectors in $V$ of length $\dim V$ is a basis of $V$.

Suppose $\dim V=n$ and $v_1,\dots ,v_n$ spans $V$. The list $v_1,\dots ,v_n$ can be reduced to a basis of $V$ (every spanning list contains a basis). However, every basis of $V$ has length $n$, so in this case the reduction is the trivial one, meaning that no elements are deleted from $v_1,\dots ,v_n$. Thus $v_1,\dots ,v_n$ is a basis of $V$, as desired.