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1.1. Motivation and definitions

The abstract definition of a matrix is historically much more recent than the techniques developed in these notes. The motivation for the definition of a matrix, given below, comes from two different places: (1) The algebraic question of solving a system of linear equations, see Section 5 for more on this question; (2) The geometric question of describing linear transformations of $n$ -dimensional space, ${\mathbb{R}}^{n}$ , see Section 6 for more on this question.

Throughout this section, $n,~{}m$ are positive integers ( $\geq 1$ ).

Definition 1.1.1.

A matrix is a rectangular array of numbers (usually real numbers in this module). The entries in the array are called the coefficients (or elements) of the matrix. If a matrix $A$ has $n$ rows and $m$ columns then $A$ is an $n\times m$ matrix. If $m=n$ then $A$ is a square matrix. We write $a_{ij}$ for the element in the $i$ -th row and $j$ -th column, starting with the $(1,1)$ coefficient in the top left corner of the array:

A=(a_{ij})_{\begin{subarray}{c}1\leq i\leq n\\ 1\leq j\leq m\end{subarray}}=\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1m}\\ a_{21}&a_{22}&\ldots&a_{2m}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\ldots&a_{nm}\\ \end{pmatrix}\;.

We commonly use $A=(a_{ij})_{\begin{subarray}{c}1\leq i\leq n\\ 1\leq j\leq m\end{subarray}}$ , or simply $A=(a_{ij})$ as a shorthand for saying that the $(i,j)$ coefficient of $A$ is $a_{ij}$ , for all $i, j$ .

Example 1.1.2.

Here $A$ is a $2\times 3$ matrix and $B$ is a $2\times 2$ matrix:

$A=\begin{pmatrix}1&2&3\\ 4&5&6\\ \end{pmatrix}\quad\hbox{and}\quad B=\begin{bmatrix}1&5\\ 7&9\\ \end{bmatrix}.$

If $A=(a_{ij})$ , then $a_{13}=3$ and $a_{22}=5$ . Matrices may be written with either round brackets $(a_{ij})$ or square brackets $[a_{ij}]$ .

Remark 1.1.3.

All the matrices we will consider have coefficients in ${\mathbb{R}}$ , although one could also take coefficients in ${\mathbb{C}}$ or ${\mathbb{Q}}$ for instance.

Definition 1.1.4.

For any $n$ and $m$ , the set of $n\times m$ matrices is denoted $\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ . If $m=n$ , then we simply write $\operatorname{M}_{{n}}({{\mathbb{R}}})$ . We call ${\mathbb{R}}$ the set of scalars. If $m=1$ , then the $n\times 1$ matrices are column vectors and we write ${\mathbb{R}}^{n}$ instead of $\operatorname{M}_{{n\times 1}}({{\mathbb{R}}})$ . Similarly, if $n=1$ , then the $1\times m$ matrices are row vectors.

We number the coefficients from top to bottom in each column

v=\begin{pmatrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n}\end{pmatrix}\in{\mathbb{R}}^{n}\;,

and from left to right in each row

v=\begin{pmatrix}v_{1}&v_{2}&\dots&v_{m}\end{pmatrix}\in\operatorname{M}_{{1% \times m}}({{\mathbb{R}}})\quad\hbox{where $v_{i}$ is the $i$-th coefficient of $v$\;.}\quad

Remark 1.1.5.

Some authors prefer to use the notation ${\mathbb{R}}^{m}$ for row vectors, that is $\operatorname{M}_{{1\times m}}({{\mathbb{R}}})$ . In these notes, ${\mathbb{R}}^{m}$ will refer to column vectors.

Example 1.1.6.

Let us write the $3\times 3$ matrix $A=(a_{ij})$ with coefficients $a_{ij}=2i-j$ , for $1\leq i,j\leq 3$ . We calculate each coefficient:

$\begin{array}[]{ccc}a_{11}=2\cdot 1-1=1\;,&a_{12}=2\cdot 1-2=0\;,&a_{13}=2% \cdot 1-3=-1\\ a_{21}=2\cdot 2-1=3\;,&a_{22}=2\cdot 2-2=2\;,&a_{23}=2\cdot 2-3=1\\ a_{31}=2\cdot 3-1=5\;,&a_{32}=2\cdot 3-2=4\;,&a_{33}=2\cdot 3-3=3\;,\end{array}$

and so

$A=\begin{pmatrix}1&0&-1\\ 3&2&1\\ 5&4&3\end{pmatrix}.$