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\title{Introducing matrices}
\author{The Whole Pantheon}
\date{\today}         

\begin{document}

\newcommand\CC{\mathbb C} % blackboard math, for the complex numbers
\newcommand\RR{\mathbb R} % blackboard math, for the real numbers
\newcommand\ZZ{\mathbb Z} % blackboard math, for the integers
\newcommand\NN{\mathbb N} % blackboard math, for the natural numbers 0, 1, 2...

\def\qed{$\Box$}      % box  indicating end of proof.


\begin{abstract}
This is a collection of introductions to matrices taken from
mathematics texts. 

% An abstract is a short summary of the main points in the project.  It
% does not mention background material or detail.
\end{abstract}

\maketitle

Each section is the introduction to matrices from a mathematics
textbook. The goal in each case is both to tell you what a matrix is {\em
  and} to explain why you ought to care.  As you read them,
think about why the author made these choices; about what audience is
being addressed; and about what you might do yourself.

A secondary purpose of this document is to show you how to do a few
things in \LaTeX . One important thing to notice is that you should
let \LaTeX\ do the numbering (of sections, equations, definitions, and so on);
then when you insert additional equations, definitions, and so on, the
numbering will be automatically corrected.  This means also that when
you refer to section \ref{sec:colley} (in this case the selection from
Susan Colley's book) you should refer to the internal label rather than
saying ``section 4.''  You can also refer to a particular equation,
like \eqref{e:matrixdef} (van der Waerden's definition of matrix
entries). 

\vfill\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{van der Waerden's {\em Algebra}, 1966,
  pp. 69--70}\label{sec:vanderW} 

Let ${\mathfrak M}$ and ${\mathfrak N}$ be vector spaces. A {\em
  linear transformation} is a mapping ${\mathbf A}$ from ${\mathfrak
  M}$ to ${\mathfrak N}$ with the following two properties:
\begin{align} % allows separately numbered equations on several  lines
              % to be aligned at symbol[s] marked with &
{\mathbf A}({\mathbf x} + {\mathbf y}) &= {\mathbf A}{\mathbf x} +
{\mathbf A}{\mathbf y} \label{e:add}\\
{\mathbf A}({\mathbf x}c) &= ({\mathbf A}{\mathbf x})c  \label{e:scalar}
\end{align}

 From \eqref{e:add} it follows as usual that 
\begin{align}
{\mathbf A}({\mathbf x} - {\mathbf y}) &= {\mathbf A}{\mathbf x} -
{\mathbf A}{\mathbf y} \label{e:minus}\\
{\mathbf A}({\mathbf x_1} +\cdots + {\mathbf x_r}) &= {\mathbf
  A}{\mathbf x_1} + \cdots + {\mathbf A}{\mathbf x_r}. \label{e:addmany}
\end{align}

If ${\mathfrak M}$ has finite dimension $m$ and if ${\mathbf
  p_1},\ldots,{\mathbf p_m}$ form a basis, then the effect of a linear
transformation ${\mathbf A}$ on an arbitrary vector is completely
determined by its effect on the basis vectors. Let
$$ % shorthand for "\begin{equation*}", where the * means no equation
   % number.
{\mathbf x} = {\mathbf p_1}x^1 + \cdots {\mathbf p_m}x^m.
$$ % shorthand for "\end{equation*}"
Then, from \eqref{e:addmany} and \eqref{e:scalar},
\begin{equation}\label{e:lincomb}
{\mathbf y} = {\mathbf A}{\mathbf x} = ({\mathbf A}{\mathbf p_1})x^1 +
\cdots + ({\mathbf A}{\mathbf p_m})x^m.
\end{equation}

If ${\mathfrak N}$ also has finite dimension $n$, then on the left and
right in \eqref{e:lincomb} we can express the vectors $\mathbf y$ and
${\mathbf A}{\mathbf p_k}$ in terms of the basis vectors ${\mathbf
  q_1},\ldots,{\mathbf q_n}$ of ${\mathfrak N}$:
\begin{align}
{\mathbf y} &=  \sum {\mathbf q_i} y^i \\ % 
{\mathbf A}{\mathbf p_k} &= \sum {\mathbf q_i}a_k^{\ i}. \label{e:definemat}
\end{align}

It follows from \eqref{e:lincomb} on comparison of coefficients that 
\begin{equation}\label{e:matrixdef}
y^i = \sum a_k^{\ i}x^k .
\end{equation} 

The linear transformation ${\mathbf A}$ is thus determined by a {\em
  matrix} $A$, that is, by a rectangular array of $mn$ elements
$a_k^{\ i}$ of the skew field $K$:
\begin{equation*}
A = \begin{pmatrix} %% Start a matrix enclosed in
                    %% _p_arentheses. Matrix entries are separated by
                    %% &, and each row is completed by \\.
a_1^{\ 1} & a_2^{\ 1} & \cdots & a_m^{\ 1} \\
\vdots & \vdots& & \vdots \\
a_1^{\ n} & a_2^{\ n} & \cdots & a_m^{\ n} \\
\end{pmatrix}.
\end{equation*}

If the bases ${\mathbf p_1},\ldots,{\mathbf p_m}$ and ${\mathbf
  q_1},\ldots,{\mathbf q_n}$ are fixed, then each linear
transformation $\mathbf A$ determines a unique matrix $A$ and
conversely. The first index $i$ is the {\em row index} and the second
index $k$ is the {\em column index} of a matrix element $a_k^{\
  i}$. The elements of the $k$th column are by \eqref{e:definemat} the
coordinates of the vector ${\mathbf A}{\mathbf p_k}$.
\vfill\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Strang's {\em Introduction to Linear Algebra},
  2003, pp. 24--26} \label{sec:strang}

The three unknowns are $x$, $y$, and $z$. The linear equations
$A{\mathbf x} = {\mathbf b}$ are

\begin{equation}\label{e:system}
\begin{aligned} % allows several unnumbered equations with a common number to be
               % aligned at the symbols marked with &
&x &&+ &2&y &&+ &3&z &&=\ \  6 \\
2&x &&+ &5&y &&+ &2&z &&=\ \  4 \\
6&x &&- &3&y &&+ &&z &&=\ \  2 \\
\end{aligned}
\end{equation}
We look for numbers $x$, $y$, $z$ that solve all three equations at
once.

\centerline{$\vdots$}

We have three rows in the row picture and three columns in the column
picture (plus the right side). The three rows and three columns
contain nine numbers. {\em These nine numbers fill a $3$ by $3$
  matrix.} The ``coefficient matrix'' has the rows and columns that
have so far been kept separate:
$$
\text{\textbf{\textit{The coefficient matrix is }}} A = \begin{bmatrix} % for
                                % matrices in square brackets
1 & 2 & 3 \\
2 & 5 & 2 \\
6 & -3 & 1 \\
\end{bmatrix}.
$$

The capital letter $A$ stands for all nine coefficients (in this
square array). The letter ${\mathbf b}$ denotes the column vector with
components $6$, $4$, $2$. The unknown ${\mathbf x}$ is also a column
vector, with components $x$, $y$, $z$. (We use boldface because it is
a vector, ${\mathbf x}$ because it is unknown.) By rows the equations
were\dots and now by matrices they are \eqref{e:mateq}. The shorthand
is $A{\mathbf x} = {\mathbf b}$:

\begin{equation}\label{e:mateq}
\text{\textbf{\textit{Matrix equation }}} \begin{bmatrix} 
1 & 2 & 3 \\
2 & 5 & 2 \\
6 & -3 & 1 \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}  = \begin{bmatrix}
6 \\
4 \\
2 \\
\end{bmatrix}.
\end{equation}
We multiply the matrix $A$ times the unknown vector ${\mathbf x}$ to
get the right side ${\mathbf b}$.

{\em Basic question}: What does it mean to ``multiply $A$ times
${\mathbf x}$''? We can multiply by rows or by columns. Either way,
$A{\mathbf x} = {\mathbf b}$ must be a correct representation of the
three equations. You do the same nine multiplications either way.
\vfill\eject


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Herstein's {\em Topics in Algebra},
  1964 pp. 228--229}\label{sec:herstein}

Although we have been discussing linear transformations for some time,
it has always been in a detached and impersonal way; to us a linear
transformation has been a symbol (very often, $T$) which acts in a
certain way on a vector space. When one gets right down to it, outside
of the few concrete examples encountered in the problems, we have
really never come face to face with specific linear
transformations. At the same time it is clear that if one were to
pursue the subject further there would often arise the need of making
a thorough and detailed study of a given linear transformation. To
mention one precise problem, presented with a linear transformation
(and suppose, for the moment, that  we have a means of recognizing
it), how does one go about, in a ``practical'' and computable way,
finding its characteristic roots?

What we seek first is a simple notation, or, perhaps more accurately,
representation, for linear transformations. We shall accomplish this
by use of a particular basis of the vector space and by use of the
action of a linear transformation on his basis. Once this much is
achieved, by means of the operations in $A(V)$ [the linear
transformations on a vector space $V$] we can induce operations for
the symbols created, making of them an algebra. This new object,
infused with an algebraic life of its own, can be studied as a
mathematical entity having an interest by itself. This study is what
comprises the subject of {\em matrix theory}.

However, to ignore the sources of these matrices, that is to
investigate the set of symbols independently of what they represent,
can be costly, for we would be throwing away a great deal of useful
information. Instead we shall always use the interplay between the
abstract, $A(V)$, and the concrete, the matrix algebra, to obtain
information one about the other.

Let $V$ be an $n$-dimensional vector space over a field $F$ and let
$v_1,\ldots,v_n$ be a basis of $V$ over $F$. If $T \in A(V)$ then $T$
is determined on any vector as soon as we know its action on a basis
of $V$. Since $T$ maps $V$ into $V$, $v_1T$, $v_2T$,\dots,$v_nT$ must
all be in $V$. As elements of $V$ each of these is realizable in a
{\em unique} way as a linear combination of $v_1,\ldots,v_n$ over
$F$. Thus:

\begin{equation*}
\begin{split}
v_1T &= \alpha_{11}v_1 + \alpha_{12}v_2 + \cdots + \alpha_{1n}v_n \\
v_2T &= \alpha_{21}v_1 + \alpha_{22}v_2 + \cdots + \alpha_{2n}v_n \\
v_iT &= \alpha_{i1}v_1 + \alpha_{i2}v_2 + \cdots + \alpha_{in}v_n \\
\vdots & \\
v_nT &= \alpha_{n1}v_1 + \alpha_{n2}v_2 + \cdots + \alpha_{nn}v_n, \\
\end{split}
\end{equation*}
where each $\alpha_{ij}\in F$. This system of equations can be written
more compactly as 
\begin{equation*}
v_iT = \sum_{j=1}^n \alpha_{ij}v_j, \text{\ for \ } i= 1, 2, \ldots, n.
\end{equation*}

The ordered set of $n^2$ numbers $\alpha_{ij}$ in $F$ completely
describes $T$. They will serve as the means of representing $T$.

\begin{definition}
Let $V$ be an $n$-dimensional vector space over $F$ and let
$v_1,\ldots,v_n$ be a basis of $V$ over $F$. If $T \in A(V)$ then the
{\em matrix of $T$ in the basis $v_1,\ldots,v_n$}, written $m(T)$, is
$$m(T) = \begin{pmatrix} 
\alpha_{11} & \alpha_{12} & \cdots & \alpha_{1n} \\
\alpha_{21} & \alpha_{22} & \cdots & \alpha_{2n} \\
\vdots & \vdots && \vdots \\
\alpha_{n1} & \alpha_{n2} & \cdots & \alpha_{nn} \\
\end{pmatrix}
$$
where $v_iT = \sum_{j=1}^n \alpha_{ij}v_j$.
\end{definition}

A matrix, then is an ordered, square array of elements of $F$, with,
as yet, no further properties, which represents the effect of a
linear transformation on a given basis.

\vfill\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Colley's {\em Vector Calculus},
 2006 pp. 29, 50}\label{sec:colley}

\noindent A \textbf{matrix} is a rectangular array of numbers. Examples of
matrices are
$$
\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix},
\begin{bmatrix} 1 & 3 \\ 2 & 7 \\ 0 & 0 \\ \end{bmatrix}, \text{ and }
\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\ \end{bmatrix}.
$$
If a matrix has $m$ rows and $n$ columns, we call it ``$m \times n$''
(read ``$m$ by $n$''). Thus, the three matrices just mentioned are,
respectively, $2\times 3$, $3\times 2$, and $4\times 4$. To some
extent, matrices behave algebraically like vectors. We discuss some
elementary matrix algebra in \S 1.6. For now, we are mainly interested
in the notion of a \textbf{determinant}, which is a real number
associated to an $n\times n$ (square) matrix. (There is no such thing
as the determinant of a nonsquare matrix.) In fact, for the purposes
of understanding the cross product, we need only study $2\times 2$ and
$3\times 3$ determinants.

\begin{definition}\label{d:det}
Let $A$ by a $2\times 2$ or $3\times 3$ matrix. Then the
\textbf{determinant} of $A$, denoted $\det A$ or $|A|$, is the real
number computed from the individual entries of $A$ as follows:
\begin{itemize}
\item $2\times 2$ case\\[.2cm] % this adds .2cm of extra vertical space
                               % before the new line forced by \\.
If $A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$, then $|A| =
\begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} = ad  - bc$.\\[.1cm]

\item  $3\times 3$ case\\[.2cm]
If $A = \begin{bmatrix} a & b & c \\ d & e & f\\ g & h &
  i\end{bmatrix}$, then \\[.2cm]
\begin{equation*}
\begin{split}
|A| =
\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{vmatrix} &= aei +
bfg + ccdh - ceg - afh - bdi \\
&= a \begin{vmatrix} e & f\\ h & i \end{vmatrix} - b \begin{vmatrix} d
  & f\\ g & i \end{vmatrix} + c \begin{vmatrix} d & e\\ g & h \end{vmatrix}\\
\end{split}
\end{equation*}
in terms of $2\times 2$ determinants.
\end{itemize}
\end{definition}

\centerline{$\vdots$}

We had a brief glance at matrices and determinants in \S 1.4 in
connection with computation of cross products. Now it's time for
another look.

A matrix is defined in \S 1.4 as a rectangular array of numbers. To
extend the discussion, we need a good notation for matrices and their
individual entries. We used the upper case Latin alphabet to denote
entire matrices and will continue to do so. We shall also adopt the
standard convention and two sets of indices (one set for rows, the
other for columns) to identify matrix entries. Thus, the generaal
$m\times n$ matrix can be written as
$$
A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{bmatrix} =
\text{(shorthand)} (a_{ij}).
$$
The first index will {\em always} represent the row position and the
second index the column position.

\vfill\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Artin's {\em Algebra},
 1991, pp. 1-2}\label{sec:artin}

\noindent Matrices play a central role in this book. They form an
important part of the theory, and many concrete examples are based on
them. Therefore it is essential to develop facility in matrix
manipulation. Since matrices pervade much of mathematics, the
techniques needed here are sure to be useful elsewhere.

The concepts which require practice to handle are {\em matrix
  multiplication} and {\em determinants}.

\noindent Let $m$ and $n$ be positive integers. An $m\times n$ matrix
is a collection of $mn$ numbers arranged in a rectangular array:
\begin{multline}
\hskip 4.6cm \text{$n$ columns} \\
\text{$m$ rows}\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots&&
  \vdots\\ a_{m1} & \cdots & a_{mn} \\ \end{bmatrix} \\
\end{multline}
For example, $\begin{bmatrix} 2 & 1 & 0 \\ 1 & 3 & 5 \\ \end{bmatrix}$
  is a $2 \times 3$ matrix.

The numbers in a matrix are called the {\em matrix entries} and are
denoted by $a_{ij}$, where $i$, $j$ are indices (integers) with $1\le
i \le m$ and $1 \le j\le n$. The index $i$ is called the {\em row
  index}, and $j$ is the {\em column index}. So $a_{ij}$ is the entry
which appears in the $i$th row and $j$th column of the matrix:
\begin{multline*}
\hskip 5.2cm j \\
\begin{matrix}\\ i\\ \\ \\ \\ \\ \\ \end{matrix}
\begin{bmatrix} &&&\cdot&&&&&& \\ 
\cdot&\cdot&\cdot&a_{ij}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
&&&\cdot&&&&&&\\ &&&\cdot&&&&&&\\ &&&\cdot&&&&&&\\&&&\cdot&&&&&&\\
&&&\cdot&&&&&&\\ \end{bmatrix}\\
\end{multline*}
In the example above, $a_{11}=2$, $a_{13}=0$, and $a_{23}=5$. 

We usually introduce a symbol such as $A$ to denote a matrix, or we
may write it as $(a_{ij})$.

A $1 \times n$ matrix is called an $n$-dimensional {\em row
  vector}. We will drop the index $i$ when $m=1$ and write the row
vector as 
\begin{equation}\label{e:rowvec}
A = [a_1\cdots a_n],\quad \text{or as}\quad A = (a_1,\ldots,a_n).
\end{equation}


\end{document}
