《线性代数应用(第8版)》详解：实战教程与MATLAB融合

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《线性代数与应用(英文版·第8版)》是由Steven J. Leon所著，他在马萨诸塞大学达特茅斯分校担任教授，该书针对计算机技术发展背景下对线性代数教育的需求进行了深度拓展。作为一本全面的教材，它不仅注重理论的严谨性，而且强调理论与实践的紧密结合。本书的核心内容包括矩阵代数的深化，如向量积和实舒尔分解，这些新增的内容旨在帮助学生深入理解线性代数的基础概念。作者在编写时注重通过丰富的实例来阐述抽象的概念，使得理论讲解易于理解和记忆，从而激发读者的学习兴趣。章节结构设计巧妙，每节后面配备大量习题，不仅有助于学生巩固所学知识，还鼓励他们通过解决问题进行自我提升，达到举一反三的效果。书中紧密与MATLAB这一强大的数学工具相结合，每章末尾都包含基于MATLAB的编程练习，让学生能在实践中掌握线性代数的实际应用。此外，MATLAB的基础操作在附录中也有详尽介绍，以便于初学者快速入门。这样的设计使得学习者能够将理论与实际操作相结合，提高他们的技能水平。《线性代数与应用(英文版·第8版)》的出版符合当今科技时代的需求，适应了教师的教学需求和学生对于理论与实践相结合的学习模式。无论是作为课堂教材还是自学参考书，它都能提供一个全面且实用的学习平台，帮助读者在解决实际问题的过程中深化对线性代数的理解，提升他们在信息技术领域的专业素养。

CHAPTER

Matrices and Systems of Equations

Probably the most important problem in mathematics is that of solving a system of

linear equations. Well over 75 percent of all mathematical problems encountered in

scientiﬁc or industrial applications involve solving a linear system at some stage. By

using the methods of modern mathematics, it is often possible to take a sophisticated

problem and reduce it to a single system of linear equations. Linear systems arise in

applications to such areas as business, economics, sociology, ecology, demography, ge-

netics, electronics, engineering, and physics. Therefore, it seems appropriate to begin

this book with a section on linear systems.

1.1 Systems of Linear Equations

A linear equation in n unknowns is an equation of the form

+ a

+···+a

= b

where a

, a

,...,a

and b are real numbers and x

, x

,...,x

are variables. A linear

system of m equations in n unknowns is then a system of the form

+ a

+···+ a

= b

+ a

+···+ a

= b

+ a

+···+a

= b

(1)

where the a

’s and the b

’s are all real numbers. We will refer to systems of the form (1)

as m ×n linear systems. The following are examples of linear systems:

(a) x

+ 2x

= 5

+ 3x

= 8

(b) x

− x

+ x

= 2

+ x

− x

= 4

+ x

= 2

− x

= 1

= 4

2 Chapter 1 Matrices and Systems of Equations

System (a) isa2× 2 system, (b) isa2× 3 system, and (c) isa3×2 system.

By a solution of an m × n system, we mean an ordered n-tuple of numbers

, x

,...,x

) that satisﬁes all the equations of the system. For example, the ordered

pair (1, 2) is a solution of system (a), since

1 · (1) + 2 · (2) = 5

2 · (1) + 3 · (2) = 8

The ordered triple (2, 0, 0) is a solution of system (b), since

1 · (2) − 1 · (0) + 1 · (0) = 2

2 · (2) + 1 · (0) − 1 · (0) = 4

Actually, system (b) has many solutions. If α is any real number, it is easily seen

that the ordered triple (2,α,α) is a solution. However, system (c) has no solution. It

follows from the third equation that the ﬁrst coordinate of any solution would have to

be 4. Using x

= 4 in the ﬁrst two equations, we see that the second coordinate must

satisfy

4 + x

= 2

4 − x

= 1

Since there is no real number that satisﬁes both of these equations, the system has no

solution. If a linear system has no solution, we say that the system is inconsistent.If

the system has at least one solution, we say that it is consistent. Thus, system (c) is

inconsistent, while systems (a) and (b) are both consistent.

The set of all solutions of a linear system is called the solution set of the system.

If a system is inconsistent, its solution set is empty. A consistent system will have a

nonempty solution set. To solve a consistent system, we must ﬁnd its solution set.

2 × 2 Systems

Let us examine geometrically a system of the form

+ a

= b

+ a

= b

Each equation can be represented graphically as a line in the plane. The ordered pair

, x

) will be a solution of the system if and only if it lies on both lines. For example,

consider the three systems

(i) x

+ x

= 2

− x

= 2

(ii) x

+ x

= 2

+ x

= 1

(iii) x

+ x

= 2

−x

− x

=−2

The two lines in system (i) intersect at the point (2, 0). Thus, {(2, 0)}is the solution

set of (i). In system (ii) the two lines are parallel. Therefore, system (ii) is inconsistent

and hence its solution set is empty. The two equations in system (iii) both represent the

same line. Any point on this line will be a solution of the system (see Figure 1.1.1).

In general, there are three possibilities: the lines intersect at a point, they are par-

allel, or both equations represent the same line. The solution set then contains either

one, zero, or inﬁnitely many points.

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