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\begin{document}

\title{Math 242, Lecture 13}
\maketitle

\section{ Outcomes of linear systems and larger systems of equations}  

When we talked about linear systems in two variables,
it seemed like they always had a single solution.  This
isn't always the case.   

\begin{example}
Solve the system of equations
\begin{align*}
x + 2y = 4 \\
3x + 6y = 8.  
\end{align*}
What happens if in the last equation $8$ is replaced by $9$?  
\end{example}

These results can be explained by geometry, as we 
illustrate with this example.     
The set of points which satisfies one linear equation in
two variables is a line.    And we saw last time that 
an intersection point corresponds to a solution.   
If there are no intersection points or many that corresponds to 
no solutions or many.  

We can use similar techniques to understand larger
systems of equations.   

\begin{example}
Analyze the system of equations
\begin{align*}
x + 2y = 3\\
2x - y = 5 \\
3x + y = 8   
\end{align*}
and the system one obtains by replacing $8$ in the last equation by $9$.   
\end{example}

 

\section{ The basics of matrices}   

Matrices are a useful piece of notation whenever
arrays of numbers occur in a problem,  
especially linear problems.   

\begin{definition}
A matrix is a collection of numbers arranged 
(indexed) in a rectangular array.  
\end{definition}

Matrices are usually represented within
brackets, like
$
\begin{bmatrix}
1 & 5 \\
-2 & \frac{1}{2}.
\end{bmatrix}
$
To specify a number we specify its {\em row}
and {\em column}.    For example, in this matrix
the number $5$ is in the first row, second column  
or in the 1,2 position.     We call a matrix an
$n$ by $m$ matrix if it has $n$ rows and $m$ columns.  
The example above is a two by two matrix.  

We may use  matrices to represent and
efficiently solve systems of equations  
by putting both the coefficients of the system
and the values of the equations in a matrix
and mimicking our usual procedure for solving
them.   

\begin{example}
Translate into matrix notation and solve the system 
\begin{align*}
x + 2y = 2 \\
x + 3y = 5.  
\end{align*}
\end{example}

Matrix notation has uses well beyond solving
systems.   

\subsection{ Arithmetic of matrices}

Before doing involved applications of matrices, we
develop ways to manipulate them and
combine them algebraically.    
(Note: though our ultimate topics of interest
lie in section 1.5 of {\em Linear Systems} packet, we
need topics from sections 1.2-1.4 to develop them).  

Matrices add by adding their entries.   

\begin{example}
$$
\begin{bmatrix}  1 &2 \\  3 & 4  \end{bmatrix}
+ \begin{bmatrix}  4 & -2 \\  3 & -5  \end{bmatrix}
= \begin{bmatrix}  5 & 0 \\  6 & -1  \end{bmatrix}.  
$$
\end{example}

Important note:   you can only add matrices of the same
size!  

Matrices also subtract in the way you would expect.  

\begin{example}
$$
\begin{bmatrix}  1 &2 & 3\\  4& 5 & 6  \end{bmatrix}
- \begin{bmatrix}  4 & 2 & -1 \\  3 & -5 & \frac{1}{2}  \end{bmatrix}
= \begin{bmatrix}  -3 & 0 & 4 \\ 1 & 10 & 5 \frac{1}{2}   \end{bmatrix}.  
$$
\end{example}

Addition and subtraction of matrices obey the same rules
as for numbers.   

The zero matrix has zero for all of its entries.     Matrices do
not change when the zero matrix is add to or subtracted from
them.  

We will see next time that the most natural way to multiply
matrices is not what you would expect.   

Some manipulations of matrices are not just translation from
those on numbers.   


\begin{definition}
The transpose of an $n$ by $m$ matrix $M$ is the $m$ by $n$ 
matrix called $M^T$ whose $i,j$th entry is the $j,i$th entry of $M$.  
\end{definition}

\begin{example}
The transpose of 
$\begin{bmatrix}  1 &2 & 3\\  4& 5 & 6  \end{bmatrix}$
is
$\begin{bmatrix}  1 & 4 \\ 2& 5 \\ 3 & 6  \end{bmatrix}$. 
\end{example}


\section{Vectors}  

%Last time we developed some basic properties
%of matrices.    
 Matrices with only one column or
row are called (column or row) {\em vectors}.   
Vectors naturally sit on a line, plane, three-space, etc.  

\begin{example}
$\begin{bmatrix}   1 \\  2 \end{bmatrix}$ and $\begin{bmatrix} 
3 & 4 & 5 
\end{bmatrix}$ are vectors, which we can represent by points
in the plane or in space.   
\end{example}

Because they are special cases of matrices, they can be
added and subtracted.     Vectors also have a fundamental
way to multiply each other.  

\begin{definition}
The dot product of a row vector $ \begin{bmatrix}  a_1 & a_2 & \cdots & a_n \end{bmatrix}$
and a column vector $\begin{bmatrix}   b_1  \\ b_2 \\ \cdots \\ b_n  \end{bmatrix} $
is the sum $a_1 b_1 + a_2 b_2 + \cdots a_n b_n$.  
\end{definition}

Note that the dot product of two vectors is not another vector but a number.  
The dot product arises in many contexts, especially geometry.   

\begin{theorem}
The dot product of $v$ and $w$ is zero if and only if 
the line between $O$ and $v$ is perpendicular to the line between
$O$ and $w$,   where $O$ is the origin - the point  $(0, \ldots, 0)$.    
\end{theorem}

\begin{example}
Verify this theorem when $v = \begin{bmatrix}  1 & 2 \end{bmatrix}$
and $w = \begin{bmatrix} 6 \\ -3 \end{bmatrix}.$   
\end{example}


\end{document}
 
 
 