61 lines
2.1 KiB
TeX
61 lines
2.1 KiB
TeX
\subsection{Review}
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\begin{align*}
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I: & cR_i, c \neq 0 \\
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II: & R_i \Leftrightarrow R_j \\\
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III: & R_i \Rightarrow R_i + cR_j
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\end{align*}
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Solving REF:
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\begin{enumerate}
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\item Set var -> non-pivot cols as params
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\item Solve var -> pivot cols backwards
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\# of nonzero rows = \# pivot pts = \# of pivot cols
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\end{enumerate}
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Gaussian Elimination
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\begin{enumerate}
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\item Given a matrix $A$, find left most non-zero \textbf{column}. If the leading number is NOT zero, use $II$ to swap rows.
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\item Ensure the rest of the column is 0 (by subtracting the current row from tht other rows)
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\item Cover the top row and continue for next rows
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\end{enumerate}
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\subsection{Consistency}
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\begin{defn}[Consistency]\ \\
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Suppose that $A$ is the Augmented Matrix of a linear system, and $R$ is a row-echelon form of $A$.
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\begin{itemize}
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\item When the system has no solution(inconsistent)?
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\subitem There is a row in $R$ with the form $(0 0 ... 0 | \otimes)$ where $\otimes \neq 0$
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\subitem Or, the last column is a pivot column
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\item When the system has exactly one solution?
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\subitem Last column is non-pivot
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\subitem All other columns are pivot columns
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\item When the system has infinitely many solutions?
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\subitem Last column is non-pivot
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\subitem Some other columns are non-pivot columns.
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\end{itemize}
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\end{defn}
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\begin{note} Notations\ \\
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For elementary row operations
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\begin{itemize}
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\item Multiply $i$th row by (nonzero) const $k$: $kR_i$
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\item Interchange $i$th and $j$th rows: $R_i \leftrightarrow R_j$
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\item Add $K$ times $i$th row to $j$th row: $R_j + kR_i$
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\end{itemize}
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\textbf{Note}
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\begin{itemize}
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\item $R_1 + R_2$ means "add 2nd row to the 1st row".
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\item $R_2 + R_1$ means "add 1nd row to the 2st row".
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\end{itemize}
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\textbf{Example}
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$$ \begin{pmatrix} a \\ b \end{pmatrix}
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\xrightarrow{R_1 + R_2} \begin{pmatrix} a + b \\ b \end{pmatrix}
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\xrightarrow{R_2 + (-1)R_1} \begin{pmatrix} a + b \\ -a \end{pmatrix}
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\xrightarrow{R_1 + R_2} \begin{pmatrix} b \\ -a \end{pmatrix}
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\xrightarrow{(-1)R_2}\begin{pmatrix} b \\ a \end{pmatrix}
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$$
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\end{note}
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