Once a basis is chosen in each space, every vector becomes a list of numbers and every linear map becomes a matrix. The matrix works on coordinates exactly as the map works on vectors, and a different choice of bases gives a different matrix for the same map.

A recipe can be written in grams or in cups. The dish is the same; the numbers on the card depend on the units chosen, and converting the card is arithmetic.

Let $T:V\to W$ be linear, $B=(\mathbf{b}_1,\ldots,\mathbf{b}_n)$ a basis of $V$ and $C$ a basis of $W$. The matrix of $T$ relative to $B$ and $C$, written $[T]_{C\leftarrow B}$, has as its column $j$ the coordinates $[T\mathbf{b}_j]_C$. When $V=W$ and one basis $B$ is used on both sides it is written $[T]_B$. With the standard bases of $\mathbb{R}^n$ and $\mathbb{R}^m$ it is the matrix of the chapter on matrices.

• $[T]_{C\leftarrow B}$ is $\dim W\times\dim V$: one column per basis vector of the domain, one row per basis vector of the codomain.
• For $$ from to , with the bases and : , , , so the matrix is .

Write $\mathbf{v}=c_1\mathbf{b}_1+\cdots+c_n\mathbf{b}_n$, so that . Linearity gives , and coordinates respect the operations, so . That is a combination of the columns of the matrix with the weights , which is the matrix times .

The coordinates of $\mathbf{v}$ are the weights $c_j$.

Linearity carries $T$ through the combination.

Coordinates respect addition and scaling.

A combination of the columns with the weights $c_j$ is the matrix times the column of weights.

Apply $T$ to each basis vector of the domain, write the result in coordinates of the codomain's basis, and set those coordinates down as the columns, in order.

The columns are $[D(1)]=(0,0)$, $[D(x)]=(1,0)$ and .

• Column $j$ of $[T]_{C\leftarrow B}$ is $[T\mathbf{b}_j]_C$.

Great work! You now know how to:

• build the matrix of a linear map relative to chosen bases;
• compute a map through coordinates;
• multiply matrices to compose maps;
• translate a kernel between coordinates and vectors.