Homogeneous Equation

Homogeneous Equations

Homogeneous Equations are SLE’s where the variables all come in the $a_1{x}_1,a_2{x}_2,\dots ,a_n{x}_n=0$ form (all the variables are $x_n$).

For above, all $x$’s could equal 0. That is the trivial solution. We’re looking for the non-trivial solution.

ℹ️ If a homogeneous equation has more variables than equations, it has infinitely many non-trivial solutions. (Thereom 1.3.1)

Linear Combinations:

Simple Row operations on columns are simple:

The $2x+5y$ is a linear combination of $x$ and $y$. Formally, when multiples of columns are summed together. Looks like: $sx+ty$

Know:

Any linear combinations of solutions to a SLE is again a solution. Shown:

Taking $x=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]$and $y=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right]$as solutions, then $sx+ty=\left[\begin{array}{c}s{x}_1+t{y}_1\\ s{x}_2+t{y}_2\\ ⋮\\ s{x}_n+t{y}_n\end{array}\right]$are also solutions.

The following thereom is really overly complicated for a really simple concept.

ℹ️ Let $A$ be an $m\times n$ matrix of rank $r$, and consider the homogeneous system with $n$ variables with A as the coeffecient matrix. Then:

1. The system has exactly $n-r$ basic solutions (1 solution for every parameter).
2. Every solution is a linear combination of these basic solutions.

What does that mean? Simply in non math-convoluted crap;

When you boil a homo matrix down, every variable has a solution. Any other solution you get is actually just a permutation of those basic solutions.