What theory involves the study of systems of linear equations and their solutions?

The theory that involves the study of systems of linear equations and their solutions is Linear Algebra. This branch of mathematics specifically deals with vectors, vector spaces, and linear mappings between these spaces. At its core, Linear Algebra focuses on how to represent and solve systems of linear equations, both in finite-dimensional and infinite-dimensional spaces.

In Linear Algebra, various methods are used to find solutions to these systems, such as matrix representation, row reduction, and determinants. This enables the exploration of concepts such as vector addition, scalar multiplication, and linear transformations, which are essential for understanding the behavior of linear systems. The systematic approach to solving these equations, including graphical, algebraic, and numerical techniques, is a hallmark of Linear Algebra.

In contrast, the other options—Trigonometry, Real Analysis, and Complex Analysis—focus on different areas of mathematics that do not specifically address the solutions of linear equations in the same way. Trigonometry focuses on the relationships between the angles and sides of triangles, Real Analysis deals with real numbers and the study of limits, continuity, and functions, and Complex Analysis investigates functions that operate on complex numbers. Each of these mathematical branches has its own foundational principles and applications that differ from those of Linear Algebra.