What is def? — Mathematics

Definition 21.1 University Mathematics — Year 1 · Chapter 21 — Matrices

$\mathcal{M}_{n,p}(K)$ is the vector space of $n \times p$ arrays $A = (a_{ij})$ of scalars ( $i$ : row, $j$ : column), of dimension $np$ (basis: the matrices $E_{ij}$ with a single $1$ ). Given bases $\mathcal{B} = (e_1, \dots, e_p)$ of $E$ and $\mathcal{C}$ of $F$ ( $\dim F = n$ ), the matrix of $u \in \mathcal{L}(E, F)$ is the array whose $j$ -th column lists the coordinates of $u(e_j)$ in $\mathcal{C}$ :

\operatorname{Mat}_{\mathcal{B},\mathcal{C}}(u) = (a_{ij}), \qquad u(e_j) = \sum_{i=1}^{n} a_{ij}\, f_i .

The map $u \mapsto \operatorname{Mat}_{\mathcal{B},\mathcal{C}}(u)$ is an isomorphism from $\mathcal{L}(E, F)$ onto $\mathcal{M}_{n,p}(K)$ (Proposition 20.2: a linear map is exactly a choice of images of the $e_j$ ).

Examples

Example 21.2 (The derivative, as a matrix)

Let $D(P) = P'$ on $\R_3[X]$ . In the monomial basis $(1, X, X^2, X^3)$ : $D(1) = 0$ , $D(X) = 1$ , $D(X^2) = 2X$ , $D(X^3) = 3X^2$ , so

\operatorname{Mat}(D) = \begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end{pmatrix}.

In the divided basis $\bigl(1,\ X,\ \frac{X^2}2,\ \frac{X^3}6\bigr)$ , each basis vector maps to the previous one ( $D\bigl(\frac{X^k}{k!}\bigr) = \frac{X^{k-1}}{(k-1)!}$ ), and the matrix becomes the pure shift: ones on the superdiagonal, zeros elsewhere. Two morals: the matrix belongs to the pair (map, basis), not to the map alone; and a good basis makes structure visible at a glance — the shift form shows instantly that $D^4 = 0$ on $\R_3[X]$ , each power of the matrix pushing its diagonal of ones one step further out.

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Examples

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