Linear algebra
Linear algebra is a fundamental branch of mathematics that focuses on vector and matrix operations and the structure of vector spaces. It is rooted in the analysis of linear equations, which are equations characterized by variables that are not multiplied by themselves or each other, such as x + 2y + 3z = 0. The field has practical applications across various disciplines, including computer science, engineering, and computer animation, with uses ranging from modeling traffic patterns to analyzing electrical currents and structural stresses in buildings.
The historical development of linear algebra includes significant contributions from mathematicians such as Gottfried Wilhelm Leibniz, who initially explored determinants, and Carl Friedrich Gauss, who advanced methods for solving linear equations. Vectors are central to linear algebra, defined as objects having both magnitude and direction, and can be manipulated through addition, subtraction, and scalar multiplication. Linear transformations, which are functions that modify vectors, are represented using matrices—rectangular arrays of numbers that can be manipulated mathematically. Understanding linear algebra provides essential tools for tackling complex problems in multiple fields and serves as a foundation for more advanced mathematical concepts.
On this Page
Subject Terms
Linear algebra
Linear algebra is a branch of mathematics concerned with vector and matrix algebra and the theory of vector spaces, which are, in very simple terms, sets of vectors meeting certain conditions. Linear algebra emerged from the study of linear equations, including simultaneous equations. A linear equation has no variable in it that is multiplied by itself or another variable. For example, x + 2y + 3z = 0 is a linear equation but x2 + 2y + 3z = 0 is not. Simultaneous equations are two or more equations with variables that have the same value. For example, x + y = 8 and x – y = 4 are simultaneous equations because in both equations, the value of x is 6 and the value of y is 2. Linear algebra has practical applications in many fields, including computer science, engineering, and computer animation. It has been used to analyze traffic flow patterns on streets and highways and solve network problems, such as the calculation of current flow in electrical circuits. It is heavily used in structural engineering to analyze the stresses and strains put on a building. Linear algebra is even used to implement rotations and translations in computer games.
Brief History
Linear algebra emerged from the study of determinants, which were used to solve systems of linear equations. Most historians credit the German mathematician Gottfried Wilhelm Leibniz with the discovery of the theory of determinants in 1693. However, some historians believe the Japanese mathematician Seki Kōwa came up with the same theory ten years earlier.
Determinants were then forgotten until Swiss mathematician Gabriel Cramer rediscovered them in 1750. However, Cramer did not use determinants as they are used today and did not explain how he came up with his results. Carl Friedrich Gauss, known as the "Prince of Mathematics," further developed determinants and solved linear equations using what is now known as Gaussian elimination, which is an algorithm, or a set of rules, used to solve systems of three or more equations.
In 1844, Hermann Grassmann, a German linguist and mathematician, published the "Theory of Extension," which was the foundation of linear algebra as it is used today. In 1848, British mathematician James Joseph Sylvester introduced the term matrix. Arthur Cayley, an English mathematician and the leader of the British school of pure mathematics, defined matrix multiplication and inverses. Cayley also used a single letter to represent a matrix and noticed the connection between matrices and determinants.
After this, interest in linear algebra waned until the invention of the computer after World War II (1939–1945). Computers were used to quickly solve systems of linear equations. After this, use of linear algebra spread because of its many practical applications.
Vectors
Vectors are central to the study of linear algebra. A vector is an object that has both a magnitude (size) and direction. Velocity, force, and acceleration are vectors. To understand the concept, imagine a vector as a line segment. The length of the line segment is its magnitude. The direction is indicated by an arrow at one end of the line segment. The end of the line segment with the arrow is called the head. The other end of the line segment, the end without the arrow, is called the tail. Vectors can be moved. As long as they are not rotated, their magnitude and direction will remain the same. Vectors that are the same size have the same magnitude and direction. A very simple way to describe a vector is to use a boldface letter such as a or b.
Vectors can be added, subtracted, and multiplied. When they are added, they are joined by placing the tail of the second vector after the head of the first or vice versa. If vector a and vector b are added, the notation is a + b or b + a. Subtracting vectors is similar to adding them, except the direction of the first vector—the one the second is being subtracted from—is reversed. Then the two vectors are subtracted. If vector a and vector b are subtracted, the notation is a – b or b – a.
Coordinates are a more precise way to describe vectors. A vector's coordinates are similar to a point on a coordinate grid. A two-dimensional vector will have two coordinates, as in (2, 4). The coordinates (2, 4) mean that the vector is 2 points to the right from (0, 0) and 4 points up from (0, 0). A three-dimensional vector will have three coordinates, as in (2, 4, 5). Vectors can have more than three dimensions. A vector will have the same number of coordinates as dimensions.
Vectors can be multiplied, although the process is more complicated than addition or subtraction. A vector can also be multiplied with a scalar. A scalar has magnitude, like a vector, but it does not have direction. A scalar is represented by a single number and not a set of coordinates.
Transformations
Linear transformations are functions that send instructions to a set of vectors. A simple transformation might be to multiply the magnitude of each vector by two. When this happens, the magnitude of the vectors doubles, but the direction remains the same. A transformation may change a set of vectors' direction, but their magnitude would remain the same.
Linear transformations are represented by matrices, which are rectangular arrays of numbers in rows and columns enclosed by parentheses or brackets. The numbers within a matrix are called the elements or entries of the matrix. The numbers are scalars—single digits representing magnitude but not direction. The size of an array is written as m x n, where m represents the number of rows and n represents the number of columns. Therefore, a matrix with two rows and three columns is called a 2 x 3 matrix. Matrixes can be added or subtracted by adding or subtracting their elements. They can also be multiplied.
Bibliography
Blyth, T.S., and E.F. Robertson. Basic Linear Algebra. Springer, 2002.
Cherney, David, et al. Linear Algebra. U of California, Davis, 2013.
Christensen, Jeff. "A Brief History of Linear Algebra." University of Utah, Apr. 2014, www.math.utah.edu/~gustafso/s2012/2270/web-projects/christensen-HistoryLinearAlgebra.pdf. Accessed 22 May 2017.
"An Intuitive Guide to Linear Algebra." Better Explained, betterexplained.com/articles/linear-algebra-guide/. Accessed 22 May 2017.
Lay, David C., and Steven R. Lay. Linear Algebra and Its Applications. 5th ed., Pearson, 2015.
O. Knill. "Use of Linear Algebra." 2003, Harvard University, www.math.harvard.edu/archive/21b‗fall‗03/handouts/use.pdf. Accessed 22 May 2017.
Singh, Kuldeep. Linear Algebra: Step by Step. Oxford UP, 2013.
Strang, Gilbert. Introduction to Linear Algebra. 5th ed., Wellesley-Cambridge P, 2016.