A homogeneous system of linear equations is a system where the constant term in each equation is zero. In simpler terms, every equation in the system is set equal to zero.
Defining a Homogeneous System
A general system of m linear equations with n unknowns, x1, x2, ..., xn, can be represented as:
a₁₁x₁ + a₁₂x₂ + ... + a₁nxₙ = b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂nxₙ = b₂
...
aₘ₁x₁ + aₘ₂x₂ + ... + aₘnxₙ = bₘThis system is homogeneous if and only if b₁ = b₂ = ... = bm = 0. Therefore, a homogeneous system has the form:
a₁₁x₁ + a₁₂x₂ + ... + a₁nxₙ = 0
a₂₁x₁ + a₂₂x₂ + ... + a₂nxₙ = 0
...
aₘ₁x₁ + aₘ₂x₂ + ... + aₘnxₙ = 0Key Characteristics of Homogeneous Systems
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Trivial Solution: A homogeneous system always has at least one solution, called the trivial solution, where x1 = x2 = ... = xn = 0.
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Non-Trivial Solutions: The main question when dealing with homogeneous systems is whether there are any solutions other than the trivial solution, which are called non-trivial solutions.
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Solution Space: The set of all solutions to a homogeneous system forms a vector space. This vector space is a subspace of Rn, where n is the number of variables.
Example
Consider the following system of equations:
2x + 3y = 0
x - y = 0This is a homogeneous system because both equations are equal to zero. The trivial solution is x = 0 and y = 0. However, in this particular case, it's the only solution.
Contrast this with:
2x + 3y = 0
4x + 6y = 0Here, the second equation is just a multiple of the first, so there are infinitely many solutions of the form x = -(3/2)y. Hence, there are non-trivial solutions.
Importance
Homogeneous systems are important in linear algebra because their solutions form vector spaces, which have well-defined properties. Studying the non-trivial solutions of a homogeneous system provides valuable information about the relationships between variables and the structure of the underlying linear system. They appear in various contexts, including eigenvalue problems and the analysis of linear transformations.
