Homogeneous Systems

Homogeneous systems are systems of linear equations where the right-hand side of the equations are all zero. This note will cover the basics of homogeneous systems and how to solve them.

---

Introduction

Let’s look at an example of what we might call a “Homogeneous System”:

$$\begin{array}{rl}{x}_1+x_3-x_5& =0\\ x_2-2x_3+2x_5& =0\\ x_4-3x_5& =0\end{array}$$

What do we notice about this system?

• Homogeneous Reduced Echelon
• 3 “lead” variables
• 2 “face” variables
• consistent - $\infty$ many solutions

General Solution

Lets assemble a vector for this system:

$$\left(\begin{array}{c}z-w\\ -2z+2w\\ 0z+w\\ 3z+0w\\ z+0w\end{array}\right)$$

We add “0w”/“0z” instead of just nothing to show what you don’t see - this makes the next step a little easier.

What we are going to to is split up this vector into two separate vectors for each variable:

$$\left(\begin{array}{c}z-w\\ -2z+2w\\ 0z+w\\ 3z+0w\\ z+0w\end{array}\right)=z\left(\begin{array}{c}1\\ -2\\ 0\\ 3\\ 1\end{array}\right)+w\left(\begin{array}{c}-1\\ 2\\ 1\\ 0\\ 0\end{array}\right)$$

This is a Linear Combination that we discussed last class! It is the combination of the different dimensions of this vector.

This gives us a Linear Span of:

$$\left(\begin{array}{c}1\\ -2\\ 0\\ 3\\ 1\end{array}\right),\left(\begin{array}{c}-1\\ 2\\ 1\\ 0\\ 0\end{array}\right)$$

These two vectors are the BASIC SOLUTIONS of the system.

While not a full solution, this is an important thing to be able to find - MATLAB can do the rest.

---

Superposition Principle

Given a homogeneous system of linear equations:

1. $\vec{0}$ is a solution ( trivial solution )
2. If $\vec{x}$ is a solution then so is $t\vec{x}$ for any scalar $t$
3. If $\vec{x},\vec{y}$ are solutions, then so is $\vec{x}+\vec{y}$
4. If $\vec{x},\vec{y}$ are solutions then so is $t\vec{x}+s\vec{y}$ for any scalar $t,s$

Essentially:

If ${\vec{x}}_1,\dots ,{\vec{x}}_n$ are solutions, then so is $t_1{\vec{x}}_1,\dots ,t_n{\vec{x}}_n$ for any scalar $t_n$