Linear Algebra

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Linear Algebra

Linear algebra includes vector algebra, matrix algebra, and the theory of vector spaces. Linear algebra originated as the study of linear equations, including the solution of simultaneous linear equations. Simultaneous equations are two or more equations containing the same variables; they are solved if specific values of the variables can be found that make all the equations simultaneously true. An equation is linear if no variable in is multiplied by any other variable or raised to any nonzero power other than 1. Thus, the equation 3x + 2y + z = 0 is a linear equation in three variables. Such equations are called “linear” because in the two-variable case, such as y = 2x, the equation can graphs as a straight line. The equation x³ + 6y + z + 5 = 0 is not linear, because the variable x is raised to the power 3 (multiplied by itself three times); this is an example of a cubic equation. The equation 5x - xy + 6z = 7 is not linear either, because the product of two variables (xy) appears in it. In mathematical terminology, we say that linear equations are always degree 1.

Two important concepts emerge in linear algebra to help facilitate the expression and solution of systems of simultaneous linear equations. They are the vector and the matrix. Vectors correspond to directed line segments. They have both magnitude (length) and direction. Matrices are rectangular arrays of numbers. They are used in dealing with the coefficients of simultaneous equations. Using vector and matrix notation, a system of linear equations can be written, in the form of a single equation, as a matrix times a vector.

Linear algebra has a wide variety of applications. It is useful in solving network problems, such as calculating current flow in various branches of complicated electronic circuits, or analyzing traffic flow patterns on city streets and interstate highways. Linear algebra is also the basis of a process called linear programming, widely used in business to solve a variety of problems that often contain a very large number of variables.

Historical background

The collection of theorems and ideas that comprise linear algebra have come together over some four centuries, beginning in the mid 1600s. The name linear algebra, however, is relatively recent. It derives from the fact that the graph of a linear equation is a straight line. In fact the beginnings of linear algebra are rooted in the early attempts of sixteenth and seventeenth century mathematicians to develop generalized methods for solving systems of linear equations. As early as 1693, Gottfried Leibniz put forth the notion of matrices and their determinants, and in 1750, Gabriel Cramer published his rule (it bears his name today) for solving n equations in n unknowns.

The concept of a vector, however, was originally introduced in physics applications to describe quantities having both magnitude and direction, such as force and velocity. Later, the concept was blended with many of the other notions of linear algebra when mathematicians realized that vectors and one column (or one row) matrices are mathematically identical.

Finally, the theory of vector spaces grew out of work on the algebra of vectors.

Fundamental principles

An equation is only true for certain values of the variables called solutions, or roots, of the equation. When it is desired that certain values of the variables make two or more equations true simultaneously (at the same time), the equations are called simultaneous equations and the values that make them true are called solutions to the system of simultaneous equations.

The graph of a linear equation, in a rectangular coordinate system, is a straight line, hence the term linear. The graph of simultaneous linear equations is a set of lines, one corresponding to each equation. The solution to a simultaneous system of equations, if it exists, is the set of numbers that correspond to the location in space where all the lines intersect in a single point.

Vectors

Since the solution to a system of simultaneous equations, as pointed out earlier, corresponds to the point in space where their graphs intersect in a single point, and since vectors represent points in space, the solution to a set of simultaneous equations is a vector. Thus, all the variables in a system of equations can be represented by a single variable, namely a vector.

Matrices

A matrix is a rectangular array of numbers, and is often used to represent the coefficients of a set of simultaneous equations. Two or more equations are simultaneous if each time a variable appears in any of the equations, it represents the same quantity. For example, suppose the following relationship exists between the ages of a brother and two sisters: Jack is three years older than his sister Mary, and eleven years older than his sister Nancy, who is half as old as Mary. There are three separate statements here, each of which can be translated into mathematical notation, as follows:

Let: j = Jack’s age, m = Mary’s age, n = Nancy’s age.

Let: j = Jack’s age, m = Mary’s age, n = Nancy’s age.
Then: j = m + 3 (1)
j = n + 11 (2)
2 n = m (3)

This is a system of three simultaneous equations in three unknowns. Each unknown age is represented by a variable. Each time a particular variable appears in an equation, it stands for the same quantity. In order to see how the concept of a matrix enters, rewrite the above equations, using the standard rules of algebra, as:

Since a matrix is a rectangular array of numbers, the coefficients of equations (1’), (2’), and (3’) can be written in the form of a matrix, A, called the matrix of coefficients, by letting each column contain the coefficients of a given variable (j, m, and n from left to right) and each row contain the coefficients of a single equation (equations (1’), (2’), and (3’) from top to bottom. That is,

Matrix multiplication is carried out by multiplying each row in the left matrix times each column in the right matrix. thinking of the left matrix as containing a number of “row vectors” and the right matrix as containing a number of “column vectors,” matrix multiplication consists of a series of vector dot products. Row 1 times column 1 produces a term in row 1 column 1 of the product matrix, row 2 times column 1 produces a term in row 2 column 1 of the product matrix, and so on, until each row has been multiplied by each column. The product matrix has the same number of rows as the left matrix and the same number of columns as the right matrix. In order that two matrices be compatible for multiplication, the right must have the same number of rows as the left has columns. The matrix with ones on the diagonal (the diagonal of a matrix begins in the upper left corner and ends in the lower right corner) and all other elements zero, is the identity element for multiplication of matrices, usually denoted by I. Thus the inverse of a matrix A is the matrix a^-1 such that AA^-1 = I. not every matrix has an inverse, however, if a square matrix has an inverse, then A^-1A = aA^-1 = I. That is, multiplication of a square matrix by its inverse is commutative.

Just as a matrix can be thought of as a collection of vectors, a vector can be thought of as a one-column, or one-row, matrix. Thus, multiplication of a vector by a matrix is accomplished using the rules of matrix multiplication. For example, let the variables in the previous example be represented by the vector j = (j,m,n). Then the product of the coefficient matrix, A, times the vector, j, results in a three-row, one-column matrix, containing terms that correspond to the left hand side of each of equations (1’), (2’), and (3’).

Finally, by expressing the constants on the right hand side of those equations as a constant column vector, c, the three equations can be written as the single matrix equation: Aj = c. This equation can be solved using the inverse of the matrix A. That is, multiplying both sides of the equation by the inverse of A provides the solution: j = A^-1c. The general method for finding the inverse of a matrix and hence the solution to a system of equations is given by Cramer’s rule.

Applications

Applications of linear algebra have grown rapidly since the introduction of the computer. Finding the inverse of a matrix, especially one that has hundreds or thousands of rows and columns, is a task easily performed by computer in a relatively short time. Virtually any problem that can be translated into the language of linear mathematics can be solved, provided a solution exists. Linear algebra is applied to problems in transportation and communication to route traffic and information; it is used in the fields of biology, sociology, and ecology to analyze and understand huge amounts of data; it is used daily by the business and economics community to maximize profits and optimize purchasing and manufacturing

KEY TERMS

Linear equation —A linear equation is one in which no product of variables appears. The graph of a linear equation is a straight line, hence the term linear.

Matrix —A matrix is a rectangular array of numbers, such as a table, having I rows and j columns.

Simultaneous equations —When the solution to an equation satisfies two or more equations at the same time, the equations are called simultaneous equations.

Variable —A variable is a quantity that is allowed to have a changing value, or that represents an unknown quantity.

Vector —A quantity or term that can be expressed in terms of both magnitude (a number) and a direction.

procedures; and it is vital to the understanding of physics, chemistry, and all types of engineering.

See also Solution of equation.