Yes, in the context of homogeneous systems of linear equations, a non-trivial solution generally implies the existence of infinitely many solutions.
Explanation
A "non-trivial solution" refers to any solution to a system of equations where at least one variable has a value other than zero. The trivial solution is when all variables are equal to zero.
For a homogeneous system of linear equations (where all equations are set to zero), the trivial solution (all variables equal to zero) always exists. The question of interest then becomes: are there any other solutions?
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Homogeneous Systems: Consider a homogeneous system represented by the matrix equation Ax = 0, where A is a matrix of coefficients and x is a vector of variables.
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Non-Trivial Solutions and Linear Dependence: A non-trivial solution exists if and only if the columns (or rows) of the matrix A are linearly dependent. Linear dependence means that at least one column (or row) can be expressed as a linear combination of the others.
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Infinite Solutions: If a non-trivial solution exists, it implies that there are infinitely many solutions. This is because if x is a solution to Ax = 0, then c x is also a solution for any scalar c. This creates a line (or higher-dimensional subspace) of solutions passing through the origin.
Example:
Consider the following homogeneous system:
x + y = 0
2x + 2y = 0
The trivial solution is x=0, y=0. However, x=1, y=-1 is also a solution (a non-trivial solution). Because this is a homogeneous system, any multiple of this solution is also a solution. For instance, x=2, y=-2; x=-3, y=3; etc. Therefore, there are infinite solutions.
Exceptions and Nuances
While a non-trivial solution almost always implies infinitely many solutions in the context of homogeneous linear systems, it's crucial to consider the context. If the system is not homogeneous, then the existence of a non-trivial solution doesn't guarantee infinite solutions. It depends on the specific equations.
Summary
In the context of homogeneous systems of linear equations, if a non-trivial solution exists, there are infinitely many solutions. The existence of a non-trivial solution indicates linear dependence within the system, leading to a continuum of solutions.
