Consistent and Inconsistent Systems of Linear Equations
Definition
- An $m\times n$ system of linear equations is
\begin{align*} \tag{*}
a_{1 1} x_1+a_{1 2}x_2+\cdots+a_{1 n}x_n& =b_1 \\
a_{2 1} x_1+a_{2 2}x_2+\cdots+a_{2 n}x_n& =b_2 \\
a_{3 1} x_1+a_{3 2}x_2+\cdots+a_{3 n}x_n& =b_3 \\
\vdots \qquad \qquad \cdots\qquad \qquad &\vdots \\
a_{m 1} x_1+a_{m 2}x_2+\cdots+a_{m n}x_n& =b_m,
\end{align*}
where $x_1, x_2, \dots, x_n$ are unknowns (variables) and $a_{i j}, b_k$ are numbers.
Thus an $m\times n$ system of linear equations consists of $m$ equations and $n$ unknowns $x_1, x_2, \dots, x_n$. - A system of linear equations is called homogeneous if the constants $b_1, b_2, \dots, b_m$ are all zero.
- A solution of the system (*) is a sequence of numbers $s_1, s_2, \dots, s_n$ such that the substitution $x_1=s_1, x_2=s_2, \dots, x_n=s_n$ satisfies all the $m$ equations in the system (*).
- The rank of a system of linear equation is the rank of the coefficient matrix.
Summary
- For a given system of linear equations, there are only three possibilities for the solution set of the system: No solution (inconsistent), a unique solution, or infinitely many solutions.
- The possibilities for the solution set of a homogeneous system is either a unique solution or infinitely many solutions.
- If $m < n$, then an $m\times n$ system is either inconsistent or it has infinitely many solutions.
- If $m < n$, then an $m \times n$ homogeneous system has infinitely many solutions.
- If a consistent $m\times n$ system of linear equation has rank $r$, then the system has $n-r$ free variables.
=solution
Problems
-
Determine all possibilities for the solution set of the system of linear equations described below.
(a) A homogeneous system of $3$ equations in $5$ unknowns.
(b) A homogeneous system of $5$ equations in $4$ unknowns.
(c) A system of $5$ equations in $4$ unknowns.
(d) A system of $2$ equations in $3$ unknowns that has $x_1=1, x_2=-5, x_3=0$ as a solution.
(e) A homogeneous system of $4$ equations in $4$ unknowns.
(f) A homogeneous system of $3$ equations in $4$ unknowns.
(g) A homogeneous system that has $x_1=3, x_2=-2, x_3=1$ as a solution.
(h) A homogeneous system of $5$ equations in $3$ unknowns and the rank of the system is $3$.
(i) A system of $3$ equations in $2$ unknowns and the rank of the system is $2$.
(j) A homogeneous system of $4$ equations in $3$ unknowns and the rank of the system is $2$. -
Determine all possibilities for the number of solutions of each of the system of linear equations described below.
(a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.
(b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.
(The Ohio State University) - Determine all possibilities for the solution set of a homogeneous system of $2$ equations in $2$ unknowns that has a solution $x_1=1, x_2=5$.
-
For what value(s) of $a$ does the system have nontrivial solutions?
\begin{align*}
&x_1+2x_2+x_3=0\\
&-x_1-x_2+x_3=0\\
& 3x_1+4x_2+ax_3=0.
\end{align*} -
Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.
\[A= \left[\begin{array}{rrr|r}
1 & 2 & 3 & 4 \\
2 &-1 & -2 & a^2 \\
-1 & -7 & -11 & a
\end{array} \right],\] where $a$ is a real number. Determine all the values of $a$ so that the corresponding system is consistent. -
Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.
(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\] where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.
(b) $A \mathbf{x}=\mathbf{0}$, where $A$ is a non-singular matrix.
(c) A homogeneous system of $3$ equations in $4$ unknowns.
(d) $A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:
\[\begin{bmatrix}
1 & 0 & -1 & 0 \\
0 &1 & 2 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.\] (The Ohio State University)