Linearly independent set of the right size is a basis

Statement

Let $F$ be a field and let $V$ be a vector space over $F$ of finite dimension $n$. Let $v_1, \dots, v_n \in V$ be $n$ vectors. If $v_1, \dots, v_n$ are linearly independent, then they form a basis of $V$. If $v_1, \dots, v_n$ span $V$, then they form a basis of $V$.

Remarks

Thus, for exactly $\dim V$ vectors, independence and spanning are equivalent. For example, three vectors of $\R^3$ form a basis if and only if the $3 \times 3$ matrix with these columns has nonzero determinant. More generally, a family $v_1, \dots, v_k$ is linearly independent if and only if its span has dimension $k$.

Explanations

L1 human

When we have exactly as many vectors as the dimension, we only need to check one of the two basis conditions. Independence gives spanning for free, and spanning gives independence for free. This halves the work of checking a basis.