Basis

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. A basis of $V$ is a family $(b_i)_{i \in I}$ of vectors of $V$ that is linearly independent and spans $V$.

Remarks

A family $(b_i)_{i \in I}$ is a basis if and only if each $v \in V$ is equal to $\sum_{i \in I} a_i b_i$ for exactly one finitely supported family of scalars $(a_i)_{i \in I}$. The scalars $a_i$ are the coordinates of $v$ in the basis. The standard basis $e_1, \dots, e_n$ is a basis of $F^n$; the vectors $(1, 1)$ and $(1, -1)$ form another basis of $\R^2$. Every vector space has a basis, and all bases of a space have the same cardinality.

Explanations

L1 human

A basis is a set of directions that is just right: enough to reach every vector, and none of them redundant. Once we fix a basis, each vector gets a unique list of coordinates. A vector space has many bases, but all of them have the same number of vectors.