- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.13 Linear spaces of matrices
$\quad$ We have seen how matrices arise as a representation for linear transformations, but they can also be considered objects in their own right, without necessarily being connected to linear transformations. As such, they form another class of mathematical objects on which algebraic operations can be defined.
$\quad$ Let $m$ and $n$ be two positive integers, and let $I_{m,n}$ be the set of all pairs of integers $(i, j)$ such that $1 \leq i \leq m$ and $1 \leq j \leq n.$ Any function $A$ whose domain is $I_{m,n}$ is called an $m \times n$ matrix. The function value $A(i, j)$ is called the $ij$-entry or $ij$-element of the matrix and will be denoted by $a_{ij}.$ It is customary to display all the function values in a rectangular array consisting of $m$ rows and $n$ columns as follows: \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} The elements $a_{ij}$ will usually be real or complex numbers, but they may be arbitrary objects of any kind. We may also denote matrices using the more compact notation \begin{align*} A &= (a_{i, j})_{i, j = 1}^{m, n} \qquad \text{or} \qquad A = (a_{i, j}). \end{align*} If $m = n,$ the matrix is said to be a square matrix. A $1 \times n$ matrix is called a row matrix, and an $m \times 1$ matrix is called a column matrix.
$\quad$ Two functions are equal if and only if they have the same domain and take the same function value at each element of the domain. Since matrices are functions, two matrices $A = (a_{ij})$ and $B = (b_{ij})$ are equal if and only if they have the same number of rows, the same number of columns, and equal entries $a_{ij} = b_{ij}$ for each pair $(i, j).$
$\quad$ Now, we assume that the entries are numbers (real or complex), and we define addition and scalar multiplication in the same way we would for a real- or complex-valued function:
$\quad$ Definition. $\quad$ If $A = (a_{ij})$ and $B = (b_{ij})$ are two $m \times n$ matrices and if $c$ is any scalar, we define matrices $A + B$ and $cA$ as follows: \begin{align*} A + B &= (a_{ij} + b_{ij}), \qquad cA = (ca_{ij}). \end{align*} The sum is defined only when $A$ and $B$ have the same size.
$\quad$ We define the zero matrix $O$ to be the $m \times n$ matrix all of whose elements are $0.$ With these definitions, it is a straightforward exercise to verify that the collection of all $m \times n$ matrices is a linear space. We denote this linear space by $M_{m,n}.$ If the entries are real numbers, the space $M_{m,n};$ if the entries are complex, $M_{m,n}$ is a complex linear space. A basis for $M_{m,n}$ consists of $mn$ matrices having one entry equal to $1$ and the rest equal to zero.