Proving the Unique Solution of the Homogeneous System Given a Unique Solution for Non-Homogeneous System

When dealing with systems of linear equations, understanding the relationship between the non-homogeneous system Ax b and the corresponding homogeneous system Ax 0 is fundamental. If the non-homogeneous system has a unique solution, what can we deduce about the homogeneous system? Let's explore this in detail.

Step 1: Understanding the Unique Solution of (Ax b)

When the system (Ax b) has a unique solution, this implies that the matrix (A) is invertible or nonsingular. An invertible matrix possesses the property of having a full rank, which means the number of linearly independent rows or columns is equal to the number of variables or equations in the system.

Step 2: Applying Invertibility to the Homogeneous System

To prove that the homogeneous equation (Ax 0) also has a unique solution, we can proceed as follows:

Assume Existence of Solutions: Suppose (x_1) is a solution to the homogeneous equation (Ax 0). Show the Only Solution is the Trivial Solution: Suppose there exists another solution (x_2) such that (Ax_2 0). The difference (x_3 x_2 - x_1) is also a solution to the homogeneous equation, i.e., (Ax_2 - x_1 0). Since (A) is invertible, we can multiply both sides of (Ax_2 - x_1 0) by (A^{-1}), resulting in (x_2 - x_1 0), or equivalently, (x_2 x_1). Therefore, any other solution to (Ax 0) must be the same as (x_1). The only solution is the trivial solution (x 0).

This logic is based on the fact that if (Ax 0) has any non-trivial solution, it would contradict the unique solution of (Ax b).

Conclusion

From the above steps, we conclude that the homogeneous system (Ax 0) has a unique solution, which is the trivial solution (x 0). This result is significant because it connects the properties of the matrix (A) to the solutions of both homogeneous and non-homogeneous systems.

Generalization

The core of the proof lies in matrix invertibility. If (A) is a square matrix and (Ax b) has a unique solution, then (A) is invertible. Consequently, the system (Ax 0) has only the trivial solution (x 0), as shown above.

Additional Insights

Even when (A) is not square, the unique solution of (Ax 0) implies that the matrix (A) has full rank, meaning the rank of (A) equals the number of columns. However, if (A) is square, the unique solution property of (Ax 0) ensures that every system (Ax b) has a unique solution. This is a powerful result that highlights the importance of matrix rank and invertibility.

In summary, if (Ax b) has a unique solution, implying (A) is invertible, then the homogeneous equation (Ax 0) also has a unique solution, which is the trivial solution (x 0). This relationship underscores the deep connection between the properties of a matrix and the solutions to the associated linear systems.