2. Arrays

In mathematics, vectors can be expressed as either a row or column of elements and matrices are a rectangular array of elements. An individual element in the vector \(\mathbf{a}\) is identified by an index denoted using a subscript, e.g., \(a_i\). The index is the position of the element in the vector starting at 1 for the first element. An individual element in a matrix is identified by two indices denoted in a subscript, e.g., \([A]_{ij}\), the first number corresponding to the row number and the second number corresponding to the column number.

\[\begin{split}\mathbf{a} = \begin{pmatrix} a_1, & a_2, & \cdots & a_n \end{pmatrix}, \qquad \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix}, \qquad A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}.\end{split}\]

In computer programming, a vector or matrix is represented using an array. Arrays can be one-dimensional (1D) where they contain a single row or column of elements, similar to a vector, or two-dimensional (2D) similar to a matrix. It is possible to have higher dimensional arrays but this is not recommended as it can over complicate a program.