Math 2.0

definition

Matrix

Definition

Let RR be a ring, for example a field, and let m,n≥0m, n \ge 0 be integers. An m×nm \times n matrix over RR is a family A=(aij)A = (a_{ij}) of elements aij∈Ra_{ij} \in R, indexed by the pairs (i,j)(i, j) with 1≤i≤m1 \le i \le m and 1≤j≤n1 \le j \le n. The element aija_{ij} is the entry of AA in row ii and column jj. The notation for AA is a rectangular array with mm rows and nn columns: A=(a11⋯a1n⋮⋱⋮am1⋯amn).A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}. The set of all m×nm \times n matrices over RR is written Mm×n(R)M_{m \times n}(R) or Rm×nR^{m \times n}.

Remarks

Two matrices are equal when they have the same size and the same entries in each position. A 1×n1 \times n matrix is a row vector, and an m×1m \times 1 matrix is a column vector. Mathlib uses arbitrary finite index types in place of {1,…,m}\{1, \dots, m\} and {1,…,n}\{1, \dots, n\}.

Explanations

L1 human

A matrix is a table of numbers with a fixed number of rows and columns. The matrix (123456)\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} has 2 rows and 3 columns, so it is a 2×32 \times 3 matrix. We name an entry by its row first and then its column.

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Sources

  1. Web https://en.wikipedia.org/wiki/Matrix_(mathematics), lead section

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