Existence of Non-zero Solutions to Homogeneous Linear Systems: n > m and m > n Cases

Introduction

Linear algebra provides a powerful framework for understanding and solving systems of linear equations. A homogeneous linear system typically looks like Ax 0, where A is an m x n matrix. The structure of the matrix, particularly the dimensionality of the space it maps into and out of, plays a crucial role in determining whether non-zero solutions exist for such systems.

Case 1: n > m

In this case, the matrix A represents a linear map from a higher-dimensional space mathbb{R}^n to a lower-dimensional space mathbb{R}^m. Specifically, for an m x n matrix with n > m, the system Ax 0 is guaranteed to have non-zero solutions. This is because the dimensions of the domain (spanned by n basis vectors) are larger than the dimensions of the codomain (spanned by m basis vectors).

The rank-nullity theorem provides a deeper insight. According to the theorem, the nullity (dimension of the kernel) of the linear transformation plus the rank (dimension of the image) equals the dimension of the domain. In our case, the rank cannot exceed m (since the codomain has dimension m). Therefore, the nullity (number of independent vectors that map to zero) is at least n - m > 0. Consequently, there must exist non-zero vectors x such that Ax 0.

Case 2: m > n

In this scenario, the matrix A maps from a lower-dimensional space mathbb{R}^n to a higher-dimensional space mathbb{R}^m. For an m x n matrix with m > n, the system Ax 0 does not necessarily have non-zero solutions.

To illustrate, consider the example: A begin{pmatrix} 1 0 0 1 0 0 end{pmatrix}. This matrix represents an injective linear map from mathbb{R}^2 to mathbb{R}^3. Since the map is injective, no non-zero vector in mathbb{R}^2 can be mapped to zero in mathbb{R}^3. Therefore, the system Ax 0 only has the trivial solution x 0.

Again, the rank-nullity theorem confirms this. The rank of A cannot exceed n, and the nullity can be zero if the rank is maximal. This implies that, in certain cases, the null space could be trivial, leading to the only solution being the trivial solution x 0.

Conclusion

The existence of non-zero solutions to the homogeneous linear system Ax 0 depends crucially on the dimensions of the domain and codomain defined by the matrix A. In the case where m > n, non-zero solutions may exist due to the higher dimensionality of the domain. Conversely, in the case where n > m, non-zero solutions are guaranteed due to the rank-nullity theorem.

Understanding these properties is fundamental in various applications, including algebraic geometry, machine learning, and optimization problems where linear systems play a central role.