The Proof of the Unique Solution Theorem

Introduction

In the realm of mathematics, the concept of a unique solution is foundational to the study of systems of equations, linear transformations, and functional analysis. The Unique Solution Theorem asserts that for a system of linear equations, if the coefficient matrix is invertible, then the system has a unique solution. This theorem is pivotal in ensuring the consistency and solvability of equations, particularly in contexts where multiple solutions may arise. The theorem bridges the gap between algebraic structure and practical applicability, offering a rigorous framework for analyzing systems that demand a single, definitive answer.

Theorem Statement

The Unique Solution Theorem states that a system of linear equations $ A\mathbf{x} = \mathbf{b} $ has a unique solution if and only if the coefficient matrix $ A $ is invertible. Here, $ A $ is an $ m \times n $ matrix, $ \mathbf{x} $ is a vector of unknowns, and $ \mathbf{b} $ is a vector of constants. The invertibility of $ A $ is equivalent to the determinant of $ A $ being non-zero, i.e., $ \det(A) \neq 0 $. This theorem guarantees that the system is both consistent and determined, meaning there is exactly one vector $ \mathbf{x} $ that satisfies the equation for all values of $ \mathbf{b} $ in the domain of the system.

Proof Outline

To prove the Unique Solution Theorem, we must establish two key properties: the necessity of invertibility for uniqueness and the sufficiency of invertibility for uniqueness.

  1. Necessity of Invertibility:
    Suppose that the system $ A\mathbf{x} = \mathbf{b} $ has a unique solution. Then, the matrix $ A $ must be invertible. This follows from the fact that if $ A $ were singular (i.e., $ \det(A) = 0 $), the system would be either inconsistent or have infinitely many solutions. The invertibility of $ A $ ensures that the transformation from $ \mathbf{x} $ to $ \mathbf{b} $ is bijective, meaning each $ \mathbf{b} $ corresponds to exactly one $ \mathbf{x} $.

  2. Sufficiency of Invertibility:
    Conversely, if $ A $ is invertible, the system $ A\mathbf{x} = \mathbf{b} $ has a unique solution. This is because the inverse of $ A $, denoted $ A^{-1} $, exists and can be used to solve for $ \mathbf{x} $:
    $$ \mathbf{x} = A^{-1}\mathbf{b} $$
    The invertibility of $ A $ guarantees the existence of $ A^{-1} $, ensuring the solution is both unique and well-defined.

Key Steps in the Proof

The proof of the Unique Solution Theorem relies on the properties of determinants and linear transformations.

  1. Determinant Criterion:
    The determinant of $ A $ is a scalar quantity that characterizes the volume of the parallelepiped spanned by the columns of $ A $. If $ \det(A) \neq 0 $, the columns of $ A $ are linearly independent, and the transformation $ A $ is bijective. This ensures that the system $ A\mathbf{x} = \mathbf{b} $ has a unique solution for any $ \mathbf{b} $.

  2. Row Echelon Form and Uniqueness:
    When $ A $ is invertible, its row echelon form is the identity matrix, and the solution $ \mathbf{x} $ is obtained by back-substitution. The uniqueness of the solution arises from the fact that no row operations can introduce dependencies, ensuring that each equation in the system is independent and contributes uniquely to the solution.

  3. Contradiction Argument:
    Assume, for contradiction, that $ A $ is singular (i.e., $ \det(A) = 0 $). In this case, the system $ A\mathbf{x} = \mathbf{b} $ would have either no solution or infinitely many solutions. This contradicts the assumption of a unique solution, thereby proving that invertibility is necessary for uniqueness.

Applications and Implications

The Unique Solution The