# 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. |

### Related post:

- FHSU fills need for teachers in rural areas – Hays Post
- Cabinet throwing up some odd ideas – Bangkok Post
- JEE Main Exam 2020 application process begins next week, everything you need to know – Hindustan Times
- A Library Browse Leads Math’s Bill Dunham to Question the Origins of the Möbius Function – Bryn Mawr Now
- Women Seen as Vital as Science Advances – All China Women’s Federation – Women of China
- Hard work pays off for Cardinal Newman students on GCSE results day – Brighton and Hove News
- 5 places to take your class for a Maths-themed trip – School Travel Organiser
- Mathematics Software Market SWOT Analysis of Top Key Player & Forecasts To 2025 – Tribaux
- ‘We are in a math crisis’ – Samuda laments poor performance in subject area, calls on retired teachers to help – Jamaica Gleaner
- Shape-shifting sheets | Harvard John A. Paulson School of Engineering and Applied Sciences – Harvard School of Engineering and Applied Sciences