Linearly Independent List of the Right Length is a Basis

To check that a list of vectors in $V$ is a basis of $V$, we must show that the list in question satisfies two properties:

1. It must be linearly independent.
2. It must span $V$.

This result and this one show that if the list in question has the right length, then we only need to check that it satisfies one of the two properties.

Linearly independent list of the right length is a basis

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

Proof. Suppose $\dim V=n$ and $v_1,\dots ,v_n$ is linearly independent in $V$. The list $v_1,\dots ,v_n$ can be extended to a basis of $V$, since every linearly independent list extends to a basis. However, every basis of $V$ has length $n$, so in this case the extension is trivial, meaning that no elements are added to $v_1,\dots ,v_n$. Thus, $v_1,\dots ,v_n$ is a basis of $V$, as desired.