Mathematics 270 Linear Algebra I

Study Guide :: Unit 1

Systems of Linear Equations and Matrices

This unit introduces you to one of the most important topics in linear algebra, systems of linear equations, together with the main object of study throughout this course, matrices. In particular, you will learn how to solve systems of linear equations using matrices.

To illustrate how this topic can arise in real life, consider the following practical example. The owner of a clothing store needs to make a profit of exactly $800 one day and $1200 the following day just by selling 20 pairs of pants and 30 shirts the first day and 40 pairs of pants and 10 shirts the second day. Under the assumption that there are enough clients to buy the shirts and pants (i.e., enough demand), the owner of the store wants to know the price at which he/she needs to sell the shirts and pants in order to guarantee the exact revenue needed for each one of the two days.

In this scenario, there are two unknowns, namely the price of the pants, which we call $x$ , and the price of the shirts, which we call $y$ . Notice that 20 pairs of pants and 30 shirts need to be sold the first day with a revenue of $800. This fact yields the following equation

$20 x + 30 y = 800$ (1.1)

while the 40 pants and 10 shirts meant to be sold on the second day with a revenue of $1200 yields the following equation

$40 x + 10 y = 1200$ (1.2)

Since the owner wants to choose prices for the pants and the shirts so that these equations are satisfied simultaneously, he/she simply wants to solve the following “system of linear equations”:

$20 x + 30 y = 800$
$40 x + 10 y = 1200$ (1.3)

Thus, it seems that the owner needs to have some knowledge of math in order to solve this problem. Also, notice that the information in these equations can be organized in the following array:

$(\begin{array}{l} 20 & 30 & 800 \\ 40 & 10 & 1200 \end{array})$ (1.4)

This array is an example of a “matrix.” After completing the reading for this unit, you will be able to not only understand and solve this linear system, but also to express similar problems in matrix notation and use this knowledge to simplify the procedure for solving them.

Objectives

After completing Unit 1, you should be able to:

understand what a system of linear equations is;
identify the correspondence between a linear system and its augmented matrix;
find the reduced row echelon and row echelon forms of a given matrix;
solve a system of linear equations using Gauss-Jordan and Gaussian elimination;
determine whether a system of linear equations is consistent or inconsistent;
provide a geometrical interpretation of a system of linear equations in two and three dimensions;
describe the basic properties of a homogeneous linear system;
identify the size of a matrix;
compute basic matrix arithmetic such as addition, subtraction, scalar product, multiplication, transpose and trace;
determine when two matrices can be multiplied;
define the coefficient and augmented matrices of a linear system;
define the transpose and the trace of a matrix;
describe the properties of matrix arithmetic;
prove some of the properties of matrix arithmetic;
define the identity matrix $I$ and know that such a matrix plays the analogous role that the number $1$ plays in the arithmetic of real numbers; and
define a square matrix.

1.1 Systems of Linear Equations

This section is devoted to introducing the basic terminology and notation that will be used throughout the course. For example, it will become apparent after completing the corresponding assigned reading that a system of $m$ linear equations with $n$ unknowns can be written in the following form,

$\begin{array}{ccccccccc} a_{11} x_{2} & + & a_{12} x_{2} & + & \dots & + & a_{1 n} x_{n} & = & b_{1} \\ a_{21} x_{2} & + & a_{22} x_{2} & + & + & a_{2 n} x_{n} & = & b_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m 1} x_{2} & + & a_{m 2} x_{2} & + & + & a_{m n} x_{n} & = & b_{m}, \end{array}$ (1.5)

where $a_{i j}$ are real numbers for $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ . This system can, in turn, be expressed in matrix notation with its “augmented matrix” (see page 6 of the textbook):

$(\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} & b_{1} \\ a_{21} & a_{22} & \dots & a_{2 n} & b_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} & b_{m} \end{matrix})$ (1.6)

Notice that this is the “generic” way of expressing a system of linear equations using “ $a_{i j}$ ” as the notation for the coefficients that do not have a specific value.

However, note that when we encounter a particular example, the coefficients “ $a_{i j}$ ” do have a value. For example, in the problem of the pants and shirts described in the introduction to this unit, we notice that Equation (1.5) is given by Equation (1.3). Therefore, we would have two equations and two unknowns, and according to the notation in Equation (1.5), the two unknowns are given by $x_{1} = x$ and $x_{2} = y$ , and their corresponding coefficients by $a_{11} = 20$ , $a_{12} = 30$ , $a_{21} = 40$ and $a_{22} = 10$ . In other, words we could organize the system with the following “augmented matrix”:

$\begin{matrix} (\begin{matrix} a_{11} & a_{12} & b_{1} \\ a_{21} & a_{22} & b_{2} \end{matrix}) = (\begin{matrix} 20 & 30 & 800 \\ 40 & 10 & 1200 \end{matrix}) . \end{matrix}$ (1.7)

Reading Assignment

Read, and study, pages 1–8 of the textbook (to “Exercise Set 1.1”).

This reading provides not only a very basic methodology for solving a system of linear equations but also a geometrical interpretation of its solution(s). In particular, the author introduces the use of elementary row operations to solve a system of linear equations (see example on page 7) and explain geometrically how the system of linear equations given by Equation (1.5) can have one solution, infinitely many solutions or no solution at all.

After completing the reading you should be able to solve the example of the pants and shirts either by simple substitution or by using elementary row operations, and obtain the solution, which is given by $x = $ 28$ , and $y = $ 8$ . Also, you should be able to interpret this solution geometrically as the intersection of the two straight lines given by the two linear equations in Equation (1.3).

Exercises

Work on the following textbook exercises from “Exercise Set 1.1” (pp. 8–10):

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 25, 27; and
“True-False Exercises.”

1.2 Gaussian and Gauss-Jordan Elimination

In this section, Anton and Rorres explain how to solve a system of linear equations using the following two procedures:

Gauss-Jordan elimination; and
Gaussian elimination.

It is important that, after completing the reading assignment, you are able to understand that the main difference between these two systematic methods for solving linear systems is that Gauss-Jordan elimination relies on reducing the augmented matrix to a “reduced row echelon form,” whereas Gaussian elimination simply relies on reducing the augmented matrix to a “row echelon form.”

These procedures are named after the German mathematician Carl Friedrich Gauss. The German engineer Wilhelm Jordan used them in the nineteenth century. There is an interesting historical note on page 15 of the textbook.

Reading Assignment

Read, and study, pages 11–22 of the textbook (to “Exercise Set 1.2”).

The first part of the reading is devoted to explaining how a matrix can be reduced, using elementary row operations, to a:

Reduced Row Echelon Form; and
Row Echelon Form.

The second part of the reading illustrates, both theoretically and with examples, how these two reductions are applied to the augmented matrix of a system of linear equations in order to solve it.

On the one hand, if we reduce the augmented matrix to a reduced row echelon form and then solve the system for the “leading variables,” we say that we have solved the system with the method of “Gauss-Jordan elimination.”

On the other hand, if we just reduce the augmented matrix to a row echelon form and then apply “back substitution,” we say that we have solved the system with the method of “Gaussian elimination.”

Pay close attention to the system considered on page 16. Note that it is solved using:

Gauss-Jordan elimination in “Example 5”; and
Gaussian elimination in “Example 7.”

Finally, this reading introduces you to the definition and some properties of a homogeneous linear system (pages 17–19 of the textbook). The importance of studying a homogeneous linear system —defined as a particular case of the general linear system given by Equation (1.5) where $b_{1}, b_{2}, \dots, b_{m} = 0$ —will become apparent as you proceed further in the course.

Exercises

Work on the following textbook exercises from “Exercise Set 1.2” (pp. 22–25):

1, 3(a, c), 5, 11, 13, 15, 17, 25, 26, 29, 31; and
“True-False Exercises” (a, b, c, e, i).

1.3 Matrices, Their Operations and Their Algebraic Properties

As seen in earlier sections, the representation of systems of linear equations as matrices (augmented matrices) is very convenient for solving the systems. In fact, a linear system can be fully expressed in a “matrix form.” Specifically, if we consider the linear system given by Equation (1.5), it will become apparent, after studying this section, that it can be rewritten in its “matrix form” as

$A \vec{x} = \vec{b}$ , (1.8)

where

$\begin{matrix} A = (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} & b_{1} \\ a_{21} & a_{22} & \dots & a_{2 n} & b_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} & b_{m} \end{matrix}) \end{matrix}$ (1.9)

denotes the “coefficient matrix” (composed of the all the coefficients of the linear equations), and

$\vec{x} = (\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{matrix})$ ; $\vec{b} = (\begin{matrix} b_{1} \\ b_{2} \\ ⋮ \\ b_{m} \end{matrix})$ (1.10)

denote the “unknown” vector (composed of all the unknowns of the linear equations) and the “constant” vector (composed of the independent terms of the linear equations), respectively.

Thus, you can clearly recognize the importance of matrices in the context of systems of linear equations, and even though a linear system of equations is just one of the examples in which the organization of numbers in arrays (matrices) poses a mathematical advantage, it does provide a clear justification for studying matrices as independent mathematical objects.

Therefore, the main objective of this section is to introduce you to the study of matrices and their operations. Also, it will become clear in the following sections how the knowledge of matrices can provide relevant information about systems of linear equations.

Note: The textbook uses a slightly different notation for vectors. In particular, the vectors $\vec{x}$ and $\vec{b}$ in Equation (1.10) are expressed as $x$ and $b$ in the textbook (see page 34). In this Study Guide, however, the notation used for vectors is that with an arrow on top of the letter instead of that with the boldface letter.

Reading Assignment

Read, and study, pages 25–36 (to “Exercise Set 1.3”) and pages 39–43 of the textbook (to “Inverse of a Matrix”).

The pages covered in this reading introduce you to the basic terminology for matrices such as entries, equality, square matrix, main diagonal, submatrices and coefficient matrix. You will also be introduced to the definition of the basic matrix operations: addition, subtraction, scalar product, multiplication, transpose and trace. Notice that some of these operations have (and others do not have) analog operations in the arithmetic of reals numbers.

The last part of this reading (pages 39–43), after having introduced the matrix operations, is devoted to the study of the arithmetic properties of these operations, such as the commutative, associative and distribute laws. You will see, after completing the reading, that these properties, which are satisfied by the real numbers, are not necessarily satisfied by matrices.

Exercises

Work on the following textbook exercises from “Exercise Set 1.3” (pp. 36–39):

1, 3(b, d, f, g, j), 5(b, d, e, g, i, k), 7(a, c, f ), 9(b), 11(b), 13(b), 15, 23, 25, 33, 35; and
“True-False Exercises” (a, b, c, n, o);

and from “Exercise Set 1.4” (pp. 49–51):

1, 2(c, d), 51 and 55.

1.4 Applications of Linear Algebra: Ancient Applications, Traffic Flow and Chemical Equations

This section is devoted to the study of some of the simplest applications of linear algebra with the information provided so far in this course. In particular, this section starts by posing practical problems that early Egyptian, Babylonian, Greek and Indian civilizations faced. After this, you will be introduced to more contemporary applications and, specifically, to problems in traffic flow and problems of balancing chemical reactions.

Reading Assignment

Read, and study, pages 533–538 (to “Exercise Set 10.2”), pages 84–86 (to “Electrical Circuits”), and pages 88–91 (from “Balancing Chemical Equations” to “Polynomial Interpolation”) of the textbook. These three sets of readings cover simple applications of linear algebra in the context of ancient times, traffic flow and chemical equations, respectively. You will only need the knowledge obtained in this first unit of the book to fully understand these applications.

Exercises

Work on the following textbook exercises from “Exercise Set 10.2” (pp. 539–540):

1, 2, 4 and 7;

and from “Exercise Set 1.9” (pp. 94–95):

1, 3, 9, 11 and 12.