\subsection{Linear Algebra}
\begin{itemize}
  \item \textbf{Linear} The study of items/planes and objects which are flat
  \item \textbf{Algebra} Objects are not as simple as numbers
\end{itemize}

\subsection{Linear Systems \& Their Solutions}

Points on a straight line are all the points $(x, y)$ on the $xy$ plane satisfying the linear eqn: $ax + by = c$, where $a, b > 0$

\subsubsection{Linear Equation}
Linear eqn in $n$ variables (unknowns) is an eqn in the form
$$ a_1x_1 + a_2x_2 + ... + a_nx_n = b$$
where $a_1, a_2, ..., a_n, b$ are constants. 

\begin{note}
In a linear system, we don't assume that $a_1, a_2, ..., a_n$ are not all 0
\begin{itemize}
  \item If $a_1 = ... = a_n = 0$ but $b \neq 0$, it is \textbf{inconsistent}

    E.g. $0x_1 + 0x_2 = 1$

  \item If $a_1 = ... = a_n = b = 0$, it is a \textbf{zero equation}

    E.g. $0x_1 + 0x_2 = 0$
  \item Linear equation which is not a zero equation is a \textbf{nonzero equation}

    E.g. $2x_1 - 3x_2 = 4$
  \item The following are not linear equations
    \begin{itemize}
      \item $xy = 2$
      \item $\sin\theta + \cos\phi = 0.2$
      \item $x_1^2 + x_2^2 + ... + x_n^2 = 1$
      \item $x = e^y$
    \end{itemize}
\end{itemize}
\end{note}

In the $xyz$ space, linear equation $ax + by + cz = d$ where $a, b, c > 0$ represents a plane

\subsubsection{Solutions to a Linear Equation}
Let $a_1x_1 + a_2x_2 + ... + a_nx_n = b$ be a linear eqn in n variables \\
For real numbers $s_1+ s_2+ ... + s_n$, if $a_1s_1 + a_2s_2 + ... + a_ns_n = b$, then $x_1 = s_1, x_2 = s_2, x_n = s_n$ is a solution to the linear equation \\
The set of all solutions is the \textbf{solution set}\\
Expression that gives the entire solution set is the \textbf{general solution}

\textbf{Zero Equation} is satified by any values of $x_1, x_2,... x_n$

General solution is given by $(x_1, x_2, ..., x_n) = (t_1, t_2, ..., t_n)$


\subsubsection{Examples: Linear equation $4x-2y = 1$}
\begin{itemize}
  \item x can take any arbitary value, say t
  \item $x = t \Rightarrow y = 2t - \frac{1}{2}$
  \item General Solution: 
    $
    \begin{cases}
      x = t & \text{t is a parameter}\\
      y = 2t - \frac{1}{2}
    \end{cases}
    $
  \item y can take any arbitary value, say s
  \item $y = s \Rightarrow x = \frac{1}{2}s + \frac{1}{4}$
  \item General Solution: 
    $
    \begin{cases}
      y = s & \text{s is a parameter}\\
      x = \frac{1}{2}s + \frac{1}{4}
    \end{cases}
    $
\end{itemize}

\subsubsection{Example: Linear equation $x_1 - 4x_2 + 7x_3 = 5$}

\begin{itemize}
  \item $x_2$ and $x_3$ can be chosen arbitarily, $s$ and $t$
  \item $x_1 = 5 + 4s -7t$
  \item General Solution: 
    $
    \begin{cases}
      x_1 = 5 + 4s -7t \\
      x_2 = s & s, t \text{ are arbitrary parameters}\\
      x_3 = t \\
    \end{cases}
    $
\end{itemize}


\subsection{Linear System}
Linear System of m linear equations in n variables is
\begin{equation}
\begin{cases}
  a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1 \\
  a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2 \\
  \vdots \\
  a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m \\
\end{cases}
\end{equation}
where $a_{ij}, b$ are real constants and $a_{ij}$ is the coeff of $x_j$ in the $i$th equation

\begin{note} Linear Systems
  \begin{itemize}
    \item If $a_{ij}$ and $b_i$ are zero, linear system is called a \textbf{zero system}
    \item If $a_{ij}$ and $b_i$ is nonzero, linear system is called a \textbf{nonzero system}
    \item If $x_1 = s_1, x_2 = s_2, ..., x_n = s_n$ is a solution to \textbf{every equation} in the system, then its a solution to the system
    \item If every equation has a solution, there might not be a solution to the system
    \item \textbf{Consistent} if it has at least 1 solution
    \item \textbf{Inconsistent} if it has no solutions
  \end{itemize}
\end{note}


\subsubsection{Example}
\begin{equation}
\begin{cases}
a_1x + b1_y = c_1 \\
a_2x + b2_y = c_2 \\
\end{cases}
\end{equation}

where $a_1, b_1, a_2, b_2$ not all zero

In $xy$ plane, each equation represents a straight line, $L_1, L_2$

\begin{itemize}
  \item If $L_1, L_2$ are parallel, there is no solution
  \item If $L_1, L_2$ are not parallel, there is 1 solution
  \item If $L_1, L_2$ coinside(same line), there are infinitely many solution
\end{itemize}

\begin{equation}
\begin{cases}
a_1x + b1_y + c_1z = d_1 \\
a_2x + b2_y + c_2z = d_2 \\
\end{cases}
\end{equation}

where $a_1, b_1, c_1, a_2, b_2, c_2$ not all zero

In $xyz$ space, each equation represents a plane, $P_1, P_2$

\begin{itemize}
  \item If $P_1, P_2$ are parallel, there is no solution
  \item If $P_1, P_2$ are not parallel, there is $\infty$ solutions (on the straight line intersection)
  \item If $P_1, P_2$ coinside(same plane), there are infinitely many solutions
  \item Same Plane $\Leftrightarrow a_1 : a_2 = b_1 : b_2 = c_1 : c_2 = d_1: d_2$
  \item Parallel Plane $\Leftrightarrow a_1 : a_2 = b_1 : b_2 = c_1 : c_2$
  \item Intersect Plane $\Leftrightarrow a_1 : a_2, b_1 : b_2, c_1 : c_2$ are not the same
\end{itemize}

\subsection{Augmented Matrix}
$
  \begin{amatrix}{3}
    a_{11} & a_{12} & a_{1n} & b_1 \\
    a_{21} & a_{12} & a_{2n} & b_2 \\
    a_{m1} & a_{m2} & a_{mn} & b_m \\
  \end{amatrix}
$

\subsection{Elementary Row Operations}
To solve a linear system we perform operations: 
\begin{itemize}
  \item Multiply equation by nonzero constant
  \item Interchange 2 equations
  \item add a constant multiple of an equation to another
\end{itemize}

Likewise, for a augmented matrix, the operations are on the \textbf{rows} of the augmented matrix

\begin{itemize}
  \item Multiply row by nonzero constant
  \item Interchange 2 rows
  \item add a constant multiple of a row to another row
\end{itemize}

To note: all these operations are revertible