What is linear algebra?
Linear algebra is the branch of algebra dealing with the study of matrices, vectors, vector spaces, and linear equations. These are mathematical functions that occur between vectors within linearity conditions or the set of successions that are a proportional outcome of a cause.
This type of algebra is a fundamental area within mathematics, especially in the field of geometry. It lets you define objects such as lines, planes, and rotations. It is also indispensable in the field of engineering because it makes it possible to calculate, model, and compute natural phenomena.
What are the elements of linear algebra?
In a linear algebra equation, whose graphical representation is a straight line, there are a number of elements that must be taken into account to solve it:
- Vectors: straight lines that indicate a defined direction and that are projected in a certain space. They are lines with points of origin, magnitudes, directions, coordinates, and lengths. They have a rectilinear graphic representation.
- Graders: elements used to describe a phenomenon with a magnitude but no direction. They can be a real, complex, or constant numbers.
- Root: the amount multiplied by itself as many times as necessary to get another amount as a result. The goal of the root is to get the base of the power by knowing the exponent and the subradical quantity.
- Matrix: a two-dimensional set of numbers arranged in rows and columns that allow for representing coefficients present in systems of linear equations.
- Determinant: the mathematical expression that results from the application of the elements in a square matrix.
How has algebra evolved throughout history?
The introduction of linear algebra in the West dates back to the year 1637, when René Descartes develop the concept of coordinates under a geometric approach, known today as Cartesian geometry. This concept proposes representing the lines and planes through linear equations, making it possible to calculate their intersections through systems of linear equations.
German mathematician Gottfried Leibniz established the use of determinants to solve linear systems in 1693. In 1750, Swiss mathematician Gabriel Cramer used this concept to solve linear systems and develop what is now known as Cramer’s rule.
Linear algebra, as it is known today, has been developed as a succession of contributions by scientists who continue to add terms. The contributions began in the year 1843 when Irish scientist William Rowan Hamilton developed the term vector and created quaternions. Quaternions are an extension of the real numbers to which imaginary units i, j, and k are added. They are based on complex numbers, which add the imaginary unit i to the set of real numbers.
A year later, German physicist Hermann Grassmann published the book The Extension Theory, developing topics and fundamental elements of this branch of algebra. Finally, in 1848, English mathematician James Joseph Sylvester added the term matrix.
What are the main characteristics of linear algebra?
- It looks at vector spaces, that is, mathematical structures where you can add between different elements (called vectors) of a set and multiply them by real or complex numbers.
- It is based on systems of linear equations with constants (numbers) and unknown information that is represented without exponents.
- It uses letters and symbols to replace numbers in arithmetic operations; these are known as variables.
- It is called linear because the equation represents a straight line in the Cartesian plane.
- It allows us to solve problems through logical and mathematical tools that can be applied to different sciences and branches of studies, but also for day-to-day situations.