Matrix of a linear map

Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$ with bases $B = (b_1, \dots, b_n)$ and $C = (c_1, \dots, c_m)$. Let $T \colon V \to W$ be a linear map. The matrix of $T$ in the bases $B$ and $C$ is the $m \times n$ matrix $[T]_{C,B} = (a_{ij})$ defined by $$T(b_j) = \sum_{i=1}^{m} a_{ij} c_i \qquad (1 \le j \le n).$$ Equivalently, column $j$ of $[T]_{C,B}$ is the coordinate vector $[T(b_j)]_C$.

Remarks

For each $v \in V$, $[T(v)]_C = [T]_{C,B} [v]_B$. The map $T \mapsto [T]_{C,B}$ is a linear isomorphism $\Hom_F(V, W) \to M_{m \times n}(F)$. For the rotation of $\R^2$ by an angle $\theta$, in the standard basis, the matrix is $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$. Mathlib writes this map as LinearMap.toMatrix.

Explanations

L1 human

To write a linear map as a matrix, apply the map to each basis vector. Write the coordinates of each image as a column. These columns, side by side, are the matrix.