### Systems of linear equations and matrices: Systems of Linear Equations

### Homogeneous and inhomogeneous systems

Homogeneous/non-homogeneous system 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 \[\left\{\;\begin{array}{llllllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!0\\ a_{21}x_1 \!\!&+&\!\! a_{22}x_2 \!\!&+&\!\! \cdots \!\!&+&\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!0\\ \vdots &&\vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n\!\!\!\!&=&\!\!\!\!0\end{array}\right.\]

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).

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.

Add 3 times the first equation to the second. This gives \(0=3 a+b\), or \(b=-3 a\).

If #b# is equal to #-3 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 \(b=-3 a\).

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