88 lines
3.7 KiB
TeX
88 lines
3.7 KiB
TeX
\section{Systems of Linear Equations -- Computational Aspects}
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\begin{definition}
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\hfill\\
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Two systems of linear equations are called \textbf{equivalent} if they have the same solution set.
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\end{definition}
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\begin{theorem}
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\hfill\\
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Let $Ax = b$ be a system of $m$ linear equations in $n$ unknowns, and let $C$ be an invertible $m \times n$ matrix. Then the system $(CA)x = Cb$ is equivalent to $Ax = b$.
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\end{theorem}a
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\begin{corollary}
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\hfill\\
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Let $Ax = b$ be a system of $m$ linear equations in $n$ unknowns. If $(A'|b')$ is obtained from $(A|b)$ by a finite number of elementary row operations, then the system $A'x = b'$ is equivalent to the original system.
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\end{corollary}
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\begin{definition}
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\hfill\\
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A matrix is said to be in \textbf{reduced row echelon form} if the following three conditions are satisfied.
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\begin{enumerate}
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\item Any row containing a nonzero entry precedes any row in which all the entries are zero (if any).
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\item The first nonzero entry in each row is the only nonzero entry in its column.
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\item The first nonzero entry in each row is 1 and it occurs in a column to the right of the first nonzero entry in the preceding row.
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\end{enumerate}
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\end{definition}
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\begin{definition}
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\hfill\\
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The following procedure for reducing an augmented matrix to reduced row echelon form is called \textbf{Gaussian elimination}. It consists of two separate parts.
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\begin{enumerate}
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\item In the \textit{forward pass}, the augmented matrix is transformed into an upper triangular matrix in which the first nonzero entry of each row is $1$, and it occurs in a column to the right of the first nonzero entry in the preceding row.
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\item In the \textit{backward pass} or \textit{back-substitution}, the upper triangular matrix is transformed into reduced row echelon form by making the first nonzero entry of each row the only nonzero entry of its column.
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\end{enumerate}
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\end{definition}
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\begin{theorem}
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\hfill\\
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Gaussian elimination transforms any matrix into its reduced row echelon form.
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\end{theorem}
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\begin{definition}
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A solution to a system of equations of the form
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\[s = s_0 + t_1u_1 + t_2u_2 + \dots +t_{n-r}u_{n-r},\]
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where $r$ is the number of nonzero solutions in $A'$ ($r \leq m$), is called a \textbf{general solution} of the system $Ax = b$. It expresses an arbitrary solution $s$ of $Ax = b$ in terms of $n - r$ parameters.
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\end{definition}
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\begin{theorem}
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\hfill\\
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Let $Ax = b$ be a system of $r$ nonzero equations in $n$ unknowns. Suppose that $\rank{A} = \rank{A|b}$ and that $(A|b)$ is in reduced row echelon form. Then
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\begin{enumerate}
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\item $\rank{A} = r$.
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\item If the general solution obtained by the procedure above is of the form
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\[s = s_0 + t_1u_1 + t_2u_2 + \dots + t_{n-r}u_{n-r},\]
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then $\{u_1, u_2, \dots, u_{n-r}\}$ is a basis for the solution set of the corresponding homogeneous system, and $s_0$ is a solution to the original system.
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\end{enumerate}
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\end{theorem}
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\subsection*{An Interpretation of the Reduced Row Echelon Form}
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\addcontentsline{toc}{subsection}{An Interpretation of the Reduced Row Echelon Form}
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\begin{theorem}
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\hfill\\
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Let $A$ be an $m \times n$ matrix of rank $r$, where $r > 0$, and let $B$ be the reduced row echelon form of $A$. Then
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\begin{enumerate}
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\item The number of nonzero rows in $B$ is $r$.
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\item For each $i = 1, 2, \dots, r$, there is a column $b_{j_i}$ of $B$ such that $b_{j_i} = e_i$.
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\item The columns of $A$ numbered $j_1, j_2, \dots, j_r$ are linearly independent.
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\item For each $k = 1, 2, \dots, n$, if column $k$ of $B$ is $d_1e_1+d_2e_2+\dots+d_re_r$, then column $k$ of $A$ is $d_1a_{j_1} + d_2a_{j_2} + \dots + d_ra_{j_r}$.
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\end{enumerate}
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\end{theorem}
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\begin{corollary}
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\hfill\\
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The reduced row echelon form of a matrix is unique.
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\end{corollary}
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