Homogeneous Equations
Homogeneous Equations are SLE’s where the variables all come in the form (all the variables are ).
For above, all ’s could equal 0. That is the trivial solution. We’re looking for the non-trivial solution.
Linear Combinations:
Simple Row operations on columns are simple:
The is a linear combination of and . Formally, when multiples of columns are summed together. Looks like:
Know:
Any linear combinations of solutions to a SLE is again a solution. Shown:
Taking and as solutions, then are also solutions.
The following thereom is really overly complicated for a really simple concept.
ℹ️ Let be an matrix of rank , and consider the homogeneous system with variables with A as the coeffecient matrix. Then:
- The system has exactly basic solutions (1 solution for every parameter).
- 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.