If the Following System Has a Trivial Solution, Then...
Finding the conditions for a system of equations to have only the trivial solution (where all variables are zero) is a fundamental concept in linear algebra. This is crucial in various applications, from understanding the properties of matrices to solving real-world problems in engineering and physics. Let's delve into this important topic.
Understanding Trivial Solutions
A trivial solution in a homogeneous system of linear equations (a system where all constants on the right-hand side are zero) is a solution where all variables are equal to zero. For example, consider the system:
ax + by = 0
cx + dy = 0
If the only solution is x = 0 and y = 0, then the system has a trivial solution. The existence of only the trivial solution implies certain properties about the coefficient matrix.
Conditions for a Trivial Solution
The key to determining if a system has only a trivial solution lies in the determinant of the coefficient matrix. Let's represent our system in matrix form:
AX = 0
Where A is the coefficient matrix and X is the column vector of variables.
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Non-Singular Matrix (Invertible Matrix): If the determinant of matrix A (denoted as |A|) is non-zero, then the matrix is non-singular or invertible. In this case, the only solution to AX = 0 is the trivial solution X = 0. This is because a non-singular matrix has a unique inverse, allowing you to directly solve for X as X = Aβ»ΒΉ0 = 0.
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Singular Matrix: Conversely, if the determinant of A is zero (|A| = 0), then the matrix is singular and does not have an inverse. In this situation, there are infinitely many solutions, meaning the trivial solution is not unique. There will be at least one free variable, allowing non-zero values to satisfy the system.
Example:
Let's consider the system:
2x + 3y = 0
4x + 6y = 0
The coefficient matrix is:
A = | 2 3 |
| 4 6 |
The determinant of A is (26) - (34) = 0. Therefore, this system is singular and has infinitely many solutions, including the trivial solution. Notice that the second equation is simply a multiple of the first equation (multiply the first by 2). This indicates linear dependence between the equations.
Practical Implications
The concept of trivial solutions has broad applications:
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Linear Independence: A set of vectors is linearly independent if and only if the only linear combination that equals the zero vector is the trivial combination (all coefficients are zero). This is directly tied to the determinant of the matrix formed by the vectors.
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Eigenvalues and Eigenvectors: Finding eigenvalues and eigenvectors involves solving homogeneous systems of equations. The existence of non-trivial solutions is crucial in determining eigenvalues and eigenspaces.
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Stability Analysis: In systems of differential equations, the stability of equilibrium points often depends on the properties of the coefficient matrix, and specifically, whether it leads to only trivial solutions for associated linear systems.
Understanding the conditions for a system to possess only a trivial solution is fundamental to many areas of mathematics and its applications. By grasping the role of the determinant, you gain a powerful tool for analyzing linear systems and interpreting their solutions. Remember, a non-zero determinant guarantees only the trivial solution exists, while a zero determinant indicates the existence of infinitely many solutions.
