Homogenous Linear Systems

Definition: A system of $m$ linear equations of $n$ variables is said to be Homogenous if all of the constant terms (those terms without variables) are zero.

For example, consider the following system of linear equations:

(1)

\begin{align} 3x + 2y - 6z = 0 \\ 2x - y + 2z = 0 \\ 4x + 0y - 3z = 0 \end{align}

Theorem 1: If a system of linear equations is homogenous, then that system is also consistent (the system contains at least one solution).

• Proof: Consider a system of $m$ linear equations of $n$ variables, and suppose that this system of homogenous, that is, the constant terms are all zero:

(2)

\begin{align} a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = 0 \\ a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = 0 \\ \vdots \quad \quad \quad \quad \quad \\ a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = 0 \end{align}

• Then $(x_1, x_2, ..., x_n) = (0, 0, ..., 0)$ is a solution to the system since $a_{11}(0) + a_{12}(0) + ... + a_{1n}(0) = 0$, $a_{21}(0) + a_{22}(0) + ... + a_{2n}(0) = 0$, …, $a_{m1}(0) + a_{m2}(0) + ... + a_{mn}(0) = 0$. So this system contains at least one solution and is therefore consistent. $\blacksquare$

Theorem 1 guarantees that a homogenous system of linear equations is consistent and always contains the solution $(x_1, x_2, ..., x_n) = (0, 0, ..., 0)$. This solution is often rather trivial though, and hence we define it as follows:

Definition: If a system of $m$ linear equations of $n$ variables is homogenous, then the systems contains at least one solution $(x_1, x_2, ..., x_n) = (0, 0, ..., 0)$ known as the Trivial Solution.