Unique Solutions in Non-Square Systems of Linear Equations: Conditions and Examples

Linear equations play a crucial role in various fields, from engineering to economics. The nature of the system of these equations—whether it is square or non-square—dictates the number and type of solutions it can have. This article explores the conditions under which a non-square system can have a unique solution, along with practical examples and the theoretical underpinnings explaining these conditions.

Understanding Non-Square Systems

A non-square system of linear equations refers to a system where the number of equations (rows) is not equal to the number of unknowns (columns). In mathematical terms, this means the system is of the form (Amathbf{x} mathbf{b}), where (A) is an (m times n) matrix with (m eq n).

Conditions for a Unique Solution

The primary conditions for a non-square system to have a unique solution are:

1. Overdetermined Systems

When the system is overdetermined (i.e., (m > n)), the system can have a unique solution if the equations are consistent and independent. For an overdetermined system to have a solution, the extra equations must be consistent with the others. Additionally, the equations must be linearly independent to avoid contradictions.

Example: Consider the system with 3 equations and 2 unknowns:

(begin{align} x y 2 2x - y 3 x - y 1 end{align})

If these equations are consistent and independent, this system can have a unique solution.

2. Underdetermined Systems

When the system is underdetermined (i.e., (m

Theoretical Insights

A deeper understanding of the conditions for a non-square system to have a unique solution involves the null space and column space of the matrix (A).

Necessary Conditions for Unique Solutions

From a theoretical perspective, a system (Amathbf{x} mathbf{b}) has a unique solution if and only if the null space of (A) is trivial. In matrix (A) being (m times n), this condition is met if (dim(operatorname{Col}A) n). This implies that the number of columns (n) must be at least as many as the minimum number of rows required to span the column space completely.

Therefore, a necessary condition for the system to have a unique solution is that (m geq n).

Row-Reduced Echelon Form (RREF)

For the system to have a unique solution, the row-reduced echelon form (RREF) of (A) must have a pivot in every column. This means that (A) must have the form (A ER), where ({R} left[begin{array}{cccc} 1 0 cdots 0 0 1 cdots 0 vdots vdots ddots vdots 0 0 cdots 1 0 0 cdots 0 vdots 0 cdots 0 end{array}right]) with some zero rows added to the bottom, (E) is an arbitrary (m times m) invertible matrix, and (mathbf{b} E^{-1}mathbf{v}), where (mathbf{v}) has zero entries corresponding to the zero rows of (R).

Constructing Examples

To construct an example where a non-square system has a unique solution, follow these steps:

Choose an (n times n) identity matrix (R). Add some zero rows to the bottom.

Select an invertible (m times m) matrix (E).

Set (mathbf{v}) to be a vector with zero entries corresponding to the zero rows of (R).

Define (mathbf{b} E^{-1}mathbf{v}).

Set (A ER).

The resulting system (Amathbf{x} mathbf{b}) will have a unique solution.

Conclusion

While non-square systems of linear equations can have unique solutions under specific conditions, understanding these conditions is essential for both theoretical and practical applications. By ensuring the system is overdetermined, consistent, and independent, or by constructing systems with specific forms, one can guarantee a unique solution.

For more detailed and in-depth analysis, our recommendations include exploring the following:

Row reduction techniques for more complex systems. Theorems and proofs related to the null space and column space of matrices. Applications of non-square systems in real-world scenarios such as signal processing and machine learning.

By mastering these concepts, one can leverage the power of linear algebra to solve a wide range of problems with precision and efficiency.