Homogeneous Linear System
A linear system is called homogeneous when all of its constant terms are equal to zero.
$$
\begin{cases}
a_{11}x_1 + \dots + a_{1n}x_n = 0 \\
a_{21}x_1 + \dots + a_{2n}x_n = 0 \\
\vdots \\
a_{m1}x_1 + \dots + a_{mn}x_n = 0
\end{cases}
$$
Associated Homogeneous Linear Systems
Every linear system can be associated with a homogeneous linear system by setting all of its constant terms to zero.
Example
$$
\begin{cases}
2x_1 - 5x_2 + x_3 = 7 \\
4x_1 + x_2 = 16
\end{cases}
\quad \text{linear system}
$$
$$
\begin{cases}
2x_1 - 5x_2 + x_3 = 0 \\
4x_1 + x_2 = 0
\end{cases}
\quad \text{associated homogeneous linear system}
$$
The Trivial Solution
Every homogeneous linear system is consistent because it always admits the zero (trivial) solution.
$$ (x_1, x_2, x_3) = (0, 0, 0) $$
When all variables are set to zero, each equation in the system is automatically satisfied.
Example
$$
\begin{cases}
2 \cdot (0) - 5 \cdot (0) + 0 = 0 \\
4 \cdot (0) + 0 = 0
\end{cases}
\quad \text{trivial solution of the homogeneous linear system}
$$
Nontrivial Solutions
Nontrivial solutions are those solutions of a homogeneous linear system that differ from the trivial one. They indicate that the system has infinitely many solutions, corresponding to a nonzero subspace of the solution space.
Example
$$ (x_1, x_2, x_3) = (2, 1, 3) $$
Matrix Representation of a Homogeneous Linear System
A homogeneous linear system
$$
\begin{cases}
a_{11}x_1 + \dots + a_{1n}x_n = 0 \\
a_{21}x_1 + \dots + a_{2n}x_n = 0 \\
\vdots \\
a_{m1}x_1 + \dots + a_{mn}x_n = 0
\end{cases}
$$
can be written in matrix form as AX = B.
$$ A \cdot X = B $$
Here, A is the m×n coefficient matrix, X is the column vector of unknowns (x1, ..., xn), and B is the constant vector.
$$
\begin{pmatrix}
a_{11} & a_{12} & \dots & a_{1n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \dots & a_{mn}
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix}
$$
In this case, since the system is homogeneous, the constant vector B consists entirely of zeros { 0, ... , 0 }. Equivalently, the matrix equation can be expressed more compactly as AX = 0, emphasizing that all the equations are set equal to zero.
