1.4 Homogeneous Systems:
Consider this equation:
\[x+y+z=0\]
What is a simple solution to this equation?
Well, we have zero on the RHS. So, if we add nothing on the LHS, then we would also get zero. So, a simple solution is to set all the variables equal to zero. This type of equation, where we have only variable terms (things that are added) on one side and zero on the other, is called a homogeneous equation. In other words, a homogeneous equation is one where the variables add up to zero.
For example,
\[x+y=0\]
and
\[2x-7y=0\]
are homogeneous equations. But:
\[x+y+z=1\]
and
\[x-2y + 5=0\]
are not homogeneous because the variables do not add up to zero. That last equation is not homogeneous even though the RHS is zero because it is just a rewrite of \[x-2y=-5\] and so, the variables multiplied by their coefficients do not add up to zero.
Now that we know what a homogeneous equation is, a homogeneous system is one that has all homogeneous equations.
Consider these systems of equations:
\[\begin{align*} 3x +2y +z &= 0 & x+y &= 0 \\ x -y -z &= 0 & y+z &= 0 \\ -x +y +2z &= 0 & \end{align*}\] \[\begin{align*} x +y -z &= 0 & 5x &= 0 \\ -2x -3y -z &= 0 & -y &= 0 \\ 5x +y +2z &= 0 & x+y+z &= 0 \\ 5x +y +2z &= 0 & \end{align*}\]
First question, are these consistent systems? If so, do they, similarly again, have a simple solution?
All of these are indeed consistent and, they have a simple solution that has all variables equal to zero. So, we can ask, can a homogeneous system ever be inconsistent?
A homogeneous system consists of all homogeneous equations so, let us think of the general case of a homogeneous equation:
\[c_{1}x_{1} + c_{2}x_{2} + c_{3}x_{3} + c_{4}x_{4} + ... + c_{n}x_{n}= 0\]
Where the \(c\)’s are some constants and \(x\)’s are variables. If we set each \(x\) variable to \(0\), then that term equals \(0\) regardless of its constant coefficient \(c\). If we add up all these zero terms, we must get zero. So, a homogeneous system is always consistent.
Side note: Homogeneous means of the same type. It refers to the fact that all the terms are of degree \(1\), as opposed to having non-zero constants. Non-zero constants are terms of degree zero that could be written as \(cx^0\) which is the same thing as a \(c\). This naming reason is not important for our purposes. It is mentioned for those interested. But you can chose to simply think of it the following way; all the variables must have the same zero solution and so it is homogeneous.
This all zero solution is called the trivial solution because well, it is simple and is easy to tell just by looking at the system. A homogeneous system may have other solutions, but it is guaranteed to have at least the trivial solution as adding nothing of everything is clearly nothing.
Since we know by Theorem 1.1 that a consistent system either has a unique solution or infinitely many,
and we know by Theorem 1.2 that a homogeneous system is always consistent,
then by Theorem 1.2 we have the following corollaries:
A homogeneous system must have one solution or infinitely many
If a homogeneous system is found to have other than the trivial solution, it must have infinitely many solutions
If a homogeneous system is found to have a unique solution, it must be the trivial solution
We can also look at this graphically. Consider any two lines.
If we pinpoint the two lines at one point at the origin (zero point),
then the only way for the two lines to have more than one solution
(that is the intersection at the origin) is for the lines to coincide.
That has the same reasoning as to why we can only have one solution or infinitely many in Theorem 1.1 but not a finite number bigger than 1.
For example, look at the following animated graph (hit ‘Play’):
As you can see, the only way you get another intersection, beside at the origin, is when the two lines overlap. Of course by then, we get infinitely many solutions and, that again agrees with our Theorem 1.1 that we just covered last section.
We can see graphically when things coincide, or perhaps more accurately, not distinguish them. So, what does that look like analytically, that is, how do the equations look?
One obvious way to have the same graphical representation is to have the same exact equation but, the more interesting way is to have an equivalent equation.
An equivalent equation does not look exactly the same but has the same solution. For example, consider the following:
\[x = 1\] \[x + 1 = 2\] \[3x = 3\] \[3x + 1 = 4\]
These equations are equivalent because they have exactly the same solution; \(x=1\). Graphically, all these equations represent the same point; \(x=1\). If you can add to an equation, multiply it by a constant or perform a combination of these, and you can get from one equation to another, then the two equations are equivalent.
In a system of equations, an equivalent equation does not add information. Therefore, equivalent equations represent one constraint. For example, a system with three equations, and two of which are equivalent equations, will have two constraints.
These concepts are two sides of the same coin. If we are interested in looking at the number of equations, we may consider equivalent equations. If we are interested in looking at independent equations, then we consider the number of constraints by keeping non-equivalent solutions. Next section, we explore further how the number of equations, constraints, and variables affects the system of equations.