Linear concepts

Summary: Linear relationships are a fundamental concept of mathematics.

From ancient civilizations to modern societies, people use linear concepts in a multitude of ways, including statistical analysis, for advanced mathematics, and in scientific applications, many of which are designed to solve real-world problems. The fundamental idea of a linear relationship involves comparing two or more quantities which form a straight line when graphed. A basic linear relationship comparing two quantities is represented by the linear equation y = mx + b, where x and y are the quantities which vary in direct proportion to each other, m represents the slope of the line, and b represents the value where the line crosses the y-axis on the coordinate plane. This idea implies that the quantities in a linear relationship depend upon each other. One of the most common ways to represent a linear relationship is the linear equation. Linear equations are equations involving one or more unknown values, called “variables,” which are of the first degree. Linear equations are the simplest type of equation, since the unknown quantities are always raised to the power of one. Spaces of lines, such as a plane, are also the simplest type of geometric space. However, linear equations and the methods of finding the solutions become more complex as the number of unknowns increases or when solving more than one linear equation at a time. Students begin to explore linear concepts in the primary grades, and these are built upon and extended throughout high school and college.

Linear equations are used extensively in applied mathematics, particularly in modeling and representing real-world phenomena. Linear relationships are also used in advanced mathematical applications and modeling, typically by reducing nonlinear equations to linear equations or by constraining events within a set, or “system,” of linear equations. The development of general methods for solving linear equations was a slow process because of the limitations of communicating and representing the unknown quantities in linear relationships. These equations were initially solved using the elementary operations of addition, subtraction, multiplication, and division. The problems and their solutions were written using words. Algebraic methods of solving equations were not developed until a system of symbolic notation replaced the use of words. The modern practice of using variables in place of unknown values did not gain widespread use until the sixteenth century. Before that time, problems were typically written using only words or by using a limited set of symbols.

Linear Equations

The earliest known linear equations and methods of solving them are found in several ancient civilizations. These societies used linear equations and systems of linear equations to solve problems arising in everyday life, particularly based on civic and government needs. Although the historical information and records that exist from these ancient civilizations are fragmented, there exists enough evidence to show how the Babylonian, Egyptian, Chinese, and Islamic civilizations used and solved linear problems. Within the Babylonian civilization, the need for computational techniques beyond simple counting arose in areas of commerce, taxation, and construction. The Babylonians wrote their problems on clay tablets and included many examples of solving linear equations and systems of linear equations. These numerical problems were expressed rhetorically, without symbolic notation, and were provided to show the method of solution for a particular example. Reasons and explanations were not given, nor were any general methods of solution. The Egyptians were also concerned with commerce, taxation, and construction. They also described the methods used to solve linear equations arising from everyday life, such as dividing loaves of bread or a given amount of grain. They wrote their problems on papyrus, which was made from a reed plant, very few of which exist today. The most famous surviving papyri are the Rhind Papyrus and the Moscow Papyrus. One particular procedure the Egyptians devised for solving linear equations is known as the “method of false position.” This procedure began with guessing a value for the unknown and then adjusting the value until the correct result was found. This method was also used by other ancient civilizations and continued to appear in elementary algebra textbooks until the nineteenth century. The Chinese used a similar method of false position but made two guesses for the unknown rather than one guess. This method is known as the “method of double false position” and was later used by Islamic mathematicians. This approach continued to be used in Europe until the 1600s when advances in symbolic notation made solving linear equations a much more simple process. It took many years for a symbolic system to develop and allow for the development of general solutions to linear equations.

Linear Modeling

Linear relationships appear extensively in modeling applications. In statistics, simple linear regression is commonly used to model the relationship between two variables when that relationship appears to be generally linear. That is, when plotted on the coordinate plane, the data tend to cluster about a straight line. It can also be used to make predictions for situations when the value of one variable is known and the other is not. The concept of linear regression was developed by Sir Francis Galton in the late nineteenth century while investigating genetic inheritance. A description of this method was published in the article “Regression Towards Mediocrity in Hereditary Stature.” The term “regression” was actually a reference by Galton to observed effects in the data, not to the method itself, yet the statistical process of fitting a line to data still bears this name. An extension of the method, called “multiple linear regression,” is used to model relationships between several variables in n dimensions.

Many types of advanced mathematical models also rely on the use of linear relationships. For example, linear programming is used in business applications as a way to model important decisions that lead to maximum profit. This modeling is accomplished by constraining the variables, such as production costs, within a system of linear equations. Linear programming is also used in a modeling process known as “linear optimization.” This modeling process has a wide variety of applications in areas such as business, finance, engineering, and industry. Linear programming and linear optimization are based on the mathematical procedure of defining all of the related variables as linear relationships. This process was first developed in the 1940s and is one of the few mathematical applications that has a wide range of practical uses as well as a theoretical development of the mathematics.

Many definitions in mathematics rely on linear approximations. The derivative of a function of one variable at a point is the slope of the tangent line (the slope of the line that best approximates the curve at a given point). Mathematicians such as Isaac Barrow and Sir Isaac Newton made linear concepts a fundamental part of their work in the development of calculus. In higher dimensions the derivative is a linear transformation that is represented as a matrix. In geometry, a surface is defined as a space that locally looks like a plane. Georg Friedrich Bernhard Riemann defined higher dimensional spaces, now called “manifolds,” as locally looking flat and possessing shortest paths that are straight. In 1917, Albert Einstein used Riemann’s mathematics in order to present a model for the universe that was consistent with his theory of relativity .

Linear Algebra

Linear algebra is a subject that is fundamental to modern mathematics and applications. It arose from the study of coefficients of systems of linear equations, and linear concepts are fundamental in this area. For example, Arthur Cayley explored linear maps or transformations, and Giuseppe Peano was the first to give an abstract definition of the algebraic structure of linear vector spaces.

The late development of a symbolic notational system used in solving linear equations slowed the development of finding general methods for solving these equations. The first breakthrough in using algebraic techniques to solve linear equations occurred in the sixteenth century when Jacques Peletier proposed a general rule of algebra. This general rule involved setting up linear relationships as equations and finding the roots, a method still used in the teaching of algebra. With the adoption of a system of symbolic notation, the applications of linear equations continue to evolve and to be used in numerous ways, from basic equation solving to advanced mathematical techniques in both pure and applied mathematics. Linear algebra has a long history in mathematics, and linear concepts are considered one of the most important concepts in mathematics because of their appearances in so many levels of both pure and applied mathematics and in a multitude of real-world applications.

Bibliography

Bressoud, David. The Queen of the Sciences: A History of Mathematics. Chantilly, VA: Teaching Company, 2008.

Coxford, Arthur, ed. The Ideas of Algebra, K–12: 1988 Yearbook. Reston, VA: National Council of Teachers of Mathematics, 1988.

Katz, Victor. A History of Mathematics. Boston, MA: Addison-Wesley, 2009.