91Ó°ÊÓ

A homogeneous system of linear equations is one where the equation is set equal to zero, written as \( A\mathbf{x} = \mathbf{0} \).
Such systems always have at least the trivial solution, where \( \mathbf{x} = \mathbf{0} \).
This means every component of vector \( \mathbf{x} \) is zero, satisfying the equation naturally.

In the context of unique solutions, if a homogeneous system only admits this trivial solution,

• there are no other vectors that satisfy \( A\mathbf{x} = \mathbf{0} \)
• this absence of non-trivial solutions reflects back to the non-homogeneous system, \( A\mathbf{x} = \mathbf{b} \).

Therefore, if the homogeneous system lacks any non-trivial solution,
it ensures that \( A\mathbf{x} = \mathbf{b} \) has only one unique solution, as no other vectors satisfy the homogeneous system that could be added to potential solutions for the given \( \mathbf{b} \).

Matrix invertibility is a fundamental concept that determines whether a matrix has an inverse.
A matrix \( A \) is invertible—or non-singular—if it has a unique inverse;
an invertible matrix can "undo" the application of itself, akin to reversing an operation.

• For \( A \) to be invertible, its determinant must be non-zero.
• An invertible matrix means full rank, where the rank equals the number of its columns.

An invertible matrix implies that \( A\mathbf{x} = \mathbf{b} \) has precisely one solution, because the equation can be solved by applying the inverse: \( \mathbf{x} = A^{-1}\mathbf{b} \).
This characteristic directly relates to the unique solution condition for both non-homogeneous systems and homogeneous systems.

When the matrix \( A \) is invertible, the corresponding homogeneous system \( A\mathbf{x} = \mathbf{0} \) should only have the trivial solution.
Thus, matrix invertibility, unique solutions, and the nature of a homogeneous system's solutions are all interconnected.

The null space of a matrix \( A \) consists of all the vector solutions \( \mathbf{x} \) to the equation \( A\mathbf{x} = \mathbf{0} \).
It provides critical insights into the behavior of a system.
For a system of equations, the dimension of the null space, also known as the nullity, indicates how many independent solutions exist.

• If the null space contains only the zero vector, the nullity is zero, indicating a unique solution for the non-homogeneous equation \( A\mathbf{x} = \mathbf{b} \).
• A non-zero nullity would suggest additional independent solutions, implying multiple possible solutions to \( A\mathbf{x} = \mathbf{b} \).

In simpler terms, the fewer solutions in the null space, the more likely the system has a unique solution.
An empty null space (except the trivial one) means \( A \) is invertible, ensuring a single outcome for any given \( \mathbf{b} \).
The null space serves as a bridge between matrix properties and the nature of solutions, ensuring a deeper understanding of linear system behaviors.