DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases

Definition

Let $B = (b_{0}, \dots, b_{n - 1})$ be an ordered basis of $V$ . The unique coordinates $v = \sum_{j < n} x_{j} b_{j}$ form the coordinate column $[v]_{B} \in M_{n \times 1} (F)$ , whose $j$ -th entry is $x_{j}$ .

Let $C = (c_{0}, \dots, c_{m - 1})$ be an ordered basis of $W$ and let $T : V \to W$ be linear. The matrix of $T$ relative to $B$ and $C$ is the matrix $[T]_{B}^{C} \in M_{m \times n} (F)$ whose $j$ -th column is $[T (b_{j})]_{C}$ . Equivalently, if $T (b_{j}) = \sum_{i < m} t_{i j} c_{i}$ , then $([T]_{B}^{C})_{i j} = t_{i j}$ .

For an empty ordered basis, these definitions give the unique coordinate column or matrix of the corresponding zero-sized shape.

Depends on

Dependency tree · 2 levels

Sources

S. Axler, Linear Algebra Done Right, 4th ed., §3C, Definition 3.31

Coordinate columns [v]B[v]B​ and matrices [T]BC[T]BC​ of linear maps relative to ordered bases

Definition

Depends on

Dependency tree · 2 levels

Sources

Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases