Coordinate vector

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a basis of $V$. Each $v \in V$ is equal to $a_1 b_1 + \cdots + a_n b_n$ for exactly one $(a_1, \dots, a_n) \in F^n$. The coordinate vector of $v$ in the basis $B$ is $$[v]_B = (a_1, \dots, a_n)^T \in F^n.$$ The scalars $a_i$ are the coordinates of $v$ in $B$.

Remarks

The coordinates depend on the order of the basis vectors, so $B$ is an ordered basis. The map $v \mapsto [v]_B$ is a linear isomorphism $V \to F^n$, see isomorphism with the coordinate space. In $\R^2$ with $B = ((1, 1), (1, -1))$, the vector $(3, 1)$ has coordinate vector $(2, 1)^T$. A change of basis matrix converts coordinates in one basis into coordinates in another basis.

Explanations

L1 human

The coordinates of a vector tell us how much of each basis vector we need to build it. Different bases give different coordinates for the same vector. After we fix a basis, each vector is just a list of numbers.