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| A '''homogenous system of equations''' has a vector of zeros on the RHS, as in: | A '''homogenous''' system of equations has a vector of zeros on the RHS, as in: |
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| If a system is known to have only one unique solution and it is homogenous, it follows that the trivial solution ''is'' the one unique solution. | If a system has a non-zero constant on the RHS, it is called '''non-homogenous'''. If a system is known to [[LinearAlgebra/ParticularSolution#Number_of_solutions|have only one unique solution]] ''and'' it is homogenous, it follows that the trivial solution ''is'' the one unique solution. |
Trivial Solution
A homogenous system of equations always has a trivial solution.
Contents
Description
A homogenous system of equations has a vector of zeros on the RHS, as in:
┌ ┐ ┌ ┐ ┌ ┐ │ 1 2│ │ x│ │ 0│ │ 3 4│ │ y│ = │ 0│ └ ┘ └ ┘ └ ┘
Such a system always has a particular solution in the form of x=0 and y=0. This is called the trivial solution.
If a system has a non-zero constant on the RHS, it is called non-homogenous.
If a system is known to have only one unique solution and it is homogenous, it follows that the trivial solution is the one unique solution.