Homogeneous and inhomogeneous systems

Systems of linear equations and matrices: Systems of Linear Equations

Homogeneous and inhomogeneous systems

A system of $m$ linear equations with $n$ unknowns $x_1, \ldots, x_n$ can be reduced by means of elementary operations to the following basic form $\left\{\;\begin{array}{llllllllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\!\,\cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!b_1\\ a_{21}x_1 \!\!&+&\!\! a_{22}x_2 \!\!&+&\!\! \cdots \!\!&+&\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!b_2\\ \vdots &&\vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n\!\!\!\!&=&\!\!\!\!b_m\end{array}\right.$ Here, all $a_{ij}$ and $b_i$ with $1\le i\le m$ and $1\le j\le n$ are real or complex numbers. The numbers $a_{ij}$ are called the coefficients of the system. The numbers $b_i$ are called the right members of the system.

A system of linear equations in which the right-hand sides of the above basic shape are all equal to zero is called a homogeneous system; a general system is called nonhomogeneous. The system of equations that is obtained from an inhomogeneous system by replacing the right-hand sides by zero is called the associated homogeneous system. Thus, the associated homogeneous system of the above system of equations is $\left\{\;\begin{array}{llllllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!0\\ a_{21}x_1 \!\!&+&\!\! a_{22}x_2 \!\!&+&\!\! \cdots \!\!&+&\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!0\\ \vdots &&\vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n\!\!\!\!&=&\!\!\!\!0\end{array}\right.$

Consider the following system of linear equations in the unknowns $x$ , $y$ , and $z$ : $\lineqs{x-y +z &=& 9\cr x+y+z &=& 1\cr}$

This is a system of $2$ linear equations in $3$ unknowns. Instead of $x_1$ , $x_2$ , $x_3$ we have named the unknowns $x$ , $y$ , $z$ . Here, the order is somewhat less obvious. In particular, it must be indicated.

The associated homogeneous system is

$\lineqs{x-y +z &=& 0\cr x+y+z &=& 0\cr}$

Each equation is written in a form which differs only from the previously discussed basic form in that the constants are now on the right of the equal sign, whereas before they were on the left.

Moreover, in each equation of the system, the unknowns appear in the same order.

General and particular solution

The solutions of an inhomogeneous system of linear equations are of the form $p+H$ , where $p$ is a fixed solution of the nonhomogeneous system (the so-called particular solution) and $H$ runs over all solutions of the associated homogeneous system (that is, $H$ is the general solution of the associated homogeneous system).

Consider the following system of linear equations in the unknowns $x$ , $y$ , and $z$ :

$\lineqs{x-y +z &=& 9\cr x+y+z &=& 1\cr}$

The system is nonhomogeneous because the right sides are not (all) equal to $0$ .

A particular solution of this system is $\rv{x,y,z} = \rv{5,-4,0}$ .

The associated homogeneous system is

$\lineqs{x-y +z &=& 0\cr x+y+z &=& 0\cr}$

The solution of this homogeneous system is $\rv{x,y,z} = \rv{-z,0,z}$ .

According to the proposition, the (overall) solution of the nonhomogeneous system is

$\rv{x,y,z} = \rv{5,-4,0}+\rv{-z,0,z} = \rv{5-z,-4,z}$

Suppose that $p=\rv{p_1,p_2,\ldots,p_n}$ is a solution of the inhomogeneous system $\left\{\;\begin{array}{lllllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!b_1\\ a_{21}x_1 \!\!&+&\!\! a_{22}x_2 \!\!&+&\!\! \cdots \!\!&+&\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!b_2\\ \vdots &&\vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n\!\!\!\!&=&\!\!\!\!b_m\end{array}\right.$ In that case, for each $i$ with $1\le i\le m$ : $a_{i1}p_1+a_{i2}p_2+\cdots+a_{in}p_n=b_i$

If $h=\rv{h_1,h_2,\ldots,h_n}$ belongs to $H$ , then, for each $i$ with $1\le i \le m$ , $a_{i1}h_1+a_{i2}h_2+\cdots+a_{in}h_n=0$ so $\begin{array}{ll} a_{i1}(p_1+h_1)+a_{i2}(p_2+h_2)+\cdots+a_{in}(p_n+h_n) & \\ \quad = a_{i1}p_1+a_{i1}h_1+a_{i2}p_2+a_{i2}h_2+\cdots+a_{in}p_n+a_{in}h_n & \\ \quad = (a_{i1}p_1+a_{i2}p_2+\cdots+a_{in}p_n) +(a_{i1}h_1+a_{i2}h_2+\cdots+a_{in}h_n) & \\ \quad = b_i+0 & \\ \quad = b_i & \end{array}$ In other words, $p+h$ is a solution of the inhomogeneous system. This proves that each sum of a particular solution and a homogeneous solution is a solution of the original system.

Conversely, suppose that $o=\rv{o_1, o_2, \ldots, o_n}$ is an arbitrary solution of the inhomogeneous system. Then for each $i$ with $1\le i \le m$ , $a_{i1}o_1+a_{i2}o_2+\cdots+a_{in}o_n=b_i$ Note that $o=p+(o-p)$ where $p$ is the particular solution. We now show that $o-p$ to $H$ , that is to say, that $o-p$ is a solution of the associated homogeneous system. For each $i$ with $1\le i\le m$ we have $\begin{array}{ll} a_{i1}(o_1-p_1)+a_{i2}(o_2-p_2)+\cdots+a_{in}(o_n-p_n) & \\ \quad = a_{i1}o_1-a_{i1}p_1+a_{i2}o_2-a_{i2}p_2+\cdots+a_{in}o_n-a_{in}p_n & \\ \quad = (a_{i1}o_1+a_{i2}o_2+\cdots+a_{in}o_n) -(a_{i1}p_1+a_{i2}p_2+\cdots+a_{in}p_n) & \\ \quad = b_i-b_i & \\ \quad = 0 & \end{array}$ It follows that $o$ belongs to $p+H$ ; in words, each solution of the nonhomogeneous system is the sum of the particular solution $p$ and a homogeneous solution.

The theorem shows that the solutions of homogeneous systems have the special property that the sum of two solutions is again a solution. But more is true:

Linear structure of the solution of homogeneous systems of linear equations

The sum of two solutions of a homogeneous system of linear equations is also a solution of the same system.

If a solution of a homogeneous system of linear equations is multiplied by a factor, it is also a solution of the same system.

Consider again, the following homogeneous set of linear equations in the unknowns $x$ , $y$ , and $z$ :

$\lineqs{x-y +z &=& 0\cr x+y+z &=& 0\cr}$

The solution of this system is $\rv{x,y,z} = \rv{-z,0,z}$ . In particular, each solution is of the form $z\cdot \rv{-1,0,1}$ .

Addition of solutions $z_1\cdot \rv{-1,0,1}$ and $z_2\cdot \rv{-1,0,1}$ gives the solution $(z_1+z_2)\cdot \rv{-1,0,1}$ .

Multiplying the solution $z\cdot \rv{-1,0,1}$ by the scalar $\lambda$ gives the solution $(\lambda \cdot z)\cdot \rv{-1,0,1}$ .

The first statement follows from the previous theorem. After all, if there are two solutions of the homogeneous system, then we can regard the first solution as a particular solution and the second as a solution of the associated homogeneous system. The theorem then indicates that the sum is a solution of the original, in this case homogeneous system.

The second statement also makes use of the fact that the constant term is equal to $0$ . We give the proof for the case of one equation with two unknowns $x$ and $y$ . The general proof is very similar. So, we start from a linear equation of the form $a\cdot x + b\cdot y=0$ , where $a$ and $b$ are constants. If $\rv{x,y}$ is a solution and $\lambda$ constant, then we must derive that $\lambda\cdot \rv{x,y}$ , which equals $\rv{\lambda\cdot x,\lambda\cdot y}$ , is a solution of the given equation. This is indeed the case:

$a\cdot(\lambda\cdot x) + b\cdot (\lambda\cdot y)=\lambda\cdot(a\cdot x + b\cdot y)=\lambda\cdot 0=0$

Give an equation in the parameters $a$ and $b$ such that the following system of linear equations is solvable:
$\eqs{-4 x+3 y&=&a\cr 8 x-6 y &=&b\cr}$

$b=-2 a$

Add 2 times the first equation to the second. This gives $0=2 a+b$ , or $b=-2 a$ .

If $b$ is equal to $-2 a$ , then the second equation is a multiple of the first, and hence redundant. It is clear then that, the first equation has a solution.

The conclusion is that the given system has a solution if and only if $b=-2 a$ .

New example