feat: 1522 l3

ma1522/lec_03.tex

@@ -0,0 +1,60 @@
+\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}

ma1522/master.tex

@@ -32,24 +32,28 @@
 \section{Lecture 1}
 \hr
 \input{lec_01.tex}
-
 \newpage
 
 \section{Lecture 2}
 \hr
 \input{lec_02.tex}
+\newpage
 \section{Lecture 3}
 \hr
 \input{lec_03.tex}
+\newpage
 \section{Lecture 4}
 \hr
 \input{lec_04.tex}
+\newpage
 \section{Lecture 5}
 \hr
 \input{lec_05.tex}
+\newpage
 \section{Lecture 6}
 \hr
 \input{lec_06.tex}
+\newpage
 
 \section{Reference}