\subsection{Review}

\begin{align*}
  I: & cR_i, c \neq 0 \\
  II: & R_i \Leftrightarrow R_j \\\
  III: & R_i \Rightarrow R_i + cR_j
\end{align*}

Solving REF: 
\begin{enumerate}
  \item Set var -> non-pivot cols as params
  \item Solve var -> pivot cols backwards

    \# of nonzero rows = \# pivot pts = \# of pivot cols
\end{enumerate}

Gaussian Elimination
\begin{enumerate}
  \item Given a matrix $A$, find left most non-zero \textbf{column}. If the leading number is NOT zero, use $II$ to swap rows.
  \item Ensure the rest of the column is 0 (by subtracting the current row from tht other rows)
  \item Cover the top row and continue for next rows
\end{enumerate}

\subsection{Consistency}
\begin{defn}[Consistency]\ \\
  Suppose that $A$ is the Augmented Matrix of a linear system, and $R$ is a row-echelon form of $A$.
  \begin{itemize}
    \item When the system has no solution(inconsistent)?
      \subitem There is a row in $R$ with the form $(0 0 ... 0 | \otimes)$ where $\otimes \neq 0$
      \subitem Or, the last column is a pivot column
  \item When the system has exactly one solution?
    \subitem Last column is non-pivot
    \subitem All other columns are pivot columns
  \item When the system has infinitely many solutions?
    \subitem  Last column is non-pivot
    \subitem Some other columns are non-pivot columns.
  \end{itemize}
\end{defn}

\begin{note} Notations\ \\
  For elementary row operations
  \begin{itemize}
    \item Multiply $i$th row by (nonzero) const $k$: $kR_i$
    \item Interchange $i$th and $j$th rows: $R_i \leftrightarrow R_j$
    \item Add $K$ times $i$th row to $j$th row: $R_j + kR_i$
  \end{itemize}
  \textbf{Note}
  \begin{itemize}
    \item $R_1 + R_2$ means "add 2nd row to the 1st row".
    \item $R_2 + R_1$ means "add 1nd row to the 2st row".
  \end{itemize}

  \textbf{Example}
$$ \begin{pmatrix} a \\ b \end{pmatrix} 
\xrightarrow{R_1 + R_2} \begin{pmatrix} a + b \\ b \end{pmatrix} 
\xrightarrow{R_2 + (-1)R_1} \begin{pmatrix} a + b \\ -a \end{pmatrix} 
\xrightarrow{R_1 + R_2} \begin{pmatrix} b \\ -a \end{pmatrix} 
\xrightarrow{(-1)R_2}\begin{pmatrix} b \\ a \end{pmatrix} 
  $$
\end{note}
