Think of linear algebra as being fundamental to the development of abstract algebra, as your understanding of simple linear expressions makes it possible to conceptualize more complex structures. A major application of linear algebra is a system of equations, or a set of many equations involving the same variables. For example, a system with three variables represents three lines intersecting at one common point within a visual space. The point of intersection is the solution; this tells you how the planes are angled in relation to one another in a vector space.
As stated previously, abstract algebra studies algebraic structures as a whole. In addition to vector spaces, abstract algebra deals with various other mathematically derived shapes and spaces ranging from binaries -- closed operations typified by elementary arithmetic -- to more complex systems, such as rings, groups, fields and modules. What separates abstract algebra from other mathematical disciplines is a focus on structures as a whole, in lieu of the individual variables therein. For example, a student of abstract algebra might be concerned with the properties of a particular geometric shape, such as inverse and symmetrical relationships between its points.
Linear algebra is shaped by axioms, or truths common to all problems and equations. For example, you can solve a basic equation, based on the reflexive axiom: “a number is equal to itself,” and the symmetric axiom: “numbers are symmetric around the equals sign.” Similarly, Euclidean geometry is an axiomatic system defining the properties of lines and certain shapes; it provides the groundwork for understanding linear equations conceptualized within two- and three-dimensional spaces.
Abstract algebra developed when theorists began to establish simple definitions of new algebraic structures not yet defined by established axioms, or the rules that characterize linear algebra and Euclidean geometry. For example, a group is a set defined not by the specific variables therein but by particular properties that characterize their relationships to one another. A simple example of a group is integers or whole numbers not including decimals or fractions. Inverse integers, such as 4 and -4, are symmetrical around zero; when multiplied, integers will always produce another integer. A more complex group is a wallpaper group, which corresponds to symmetries in a two-dimensional geometric pattern.