What theory studies the properties and behaviors of vectors in mathematical spaces?

The study of properties and behaviors of vectors in mathematical spaces is primarily encompassed in Linear Algebra. This branch of mathematics focuses on vector spaces and the linear mappings between them, including concepts such as vector addition, scalar multiplication, linear transformations, and systems of linear equations.

In Linear Algebra, vectors are crucial entities that can be analyzed through their dimensions, coordinates, and relations to one another within various mathematical frameworks. The behavior of vectors is explored in terms of operations such as dot products, cross products, and projections, which are foundational to understanding multi-dimensional spaces.

While "Vector Theory" might informally describe the study of vectors, it does not constitute a formally recognized discipline within mathematics in the same way that Linear Algebra does. Other options like Complex Analysis and Real Analysis focus on different aspects of mathematical study, such as functions of complex variables and real-valued functions, respectively, and do not center on the study of vectors specifically. Thus, Linear Algebra is the appropriate field for examining the properties and behaviors of vectors in mathematical contexts.