85 lines
3.6 KiB
TeX
85 lines
3.6 KiB
TeX
\section{Systems of Linear Equations -- Computational Aspects}
|
|
|
|
\begin{definition}
|
|
\hfill\\
|
|
Two systems of linear equations are called \textbf{equivalent} if they have the same solution set.
|
|
\end{definition}
|
|
|
|
\begin{theorem}
|
|
\hfill\\
|
|
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 $Ax = b$.
|
|
\end{theorem}a
|
|
|
|
\begin{corollary}
|
|
\hfill\\
|
|
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.
|
|
\end{corollary}
|
|
|
|
\begin{definition}
|
|
\hfill\\
|
|
A matrix is said to be in \textbf{reduced row echelon form} if the following three conditions are satisfied.
|
|
|
|
\begin{enumerate}
|
|
\item Any row containing a nonzero entry precedes any row in which all the entries are zero (if any).
|
|
|
|
\item The first nonzero entry in each row is the only nonzero entry in its column.
|
|
|
|
\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.
|
|
\end{enumerate}
|
|
\end{definition}
|
|
|
|
\begin{definition}
|
|
\hfill\\
|
|
The following procedure for reducing an augmented matrix to reduced row echelon form is called \textbf{Gaussian elimination}. It consists of two separate parts.
|
|
|
|
\begin{enumerate}
|
|
\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.
|
|
|
|
\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.
|
|
\end{enumerate}
|
|
\end{definition}
|
|
|
|
\begin{theorem}
|
|
\hfill\\
|
|
Gaussian elimination transforms any matrix into its reduced row echelon form.
|
|
\end{theorem}
|
|
|
|
\begin{definition}
|
|
A solution to a system of equations of the form
|
|
|
|
\[s = s_0 + t_1u_1 + t_2u_2 + \dots +t_{n-r}u_{n-r},\]
|
|
|
|
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.
|
|
\end{definition}
|
|
|
|
\begin{theorem}
|
|
\hfill\\
|
|
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
|
|
|
|
\begin{enumerate}
|
|
\item $\rank{A} = r$.
|
|
\item If the general solution obtained by the procedure above is of the form
|
|
|
|
\[s = s_0 + t_1u_1 + t_2u_2 + \dots + t_{n-r}u_{n-r},\]
|
|
|
|
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.
|
|
\end{enumerate}
|
|
\end{theorem}
|
|
|
|
\begin{theorem}
|
|
\hfill\\
|
|
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
|
|
|
|
\begin{enumerate}
|
|
\item The number of nonzero rows in $B$ is $r$.
|
|
\item For each $i = 1, 2, \dots, r$, there is a column $b_{j_i}$ of $B$ such that $b_{j_i} = e_i$.
|
|
\item The columns of $A$ numbered $j_1, j_2, \dots, j_r$ are linearly independent.
|
|
\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 $d_1a_{j_1} + d_2a_{j_2} + \dots + d_ra_{j_r}$.
|
|
\end{enumerate}
|
|
\end{theorem}
|
|
|
|
\begin{corollary}
|
|
\hfill\\
|
|
The reduced row echelon form of a matrix is unique.
|
|
\end{corollary}
|