线性代数第一课程习题与解答手册

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"Exercise and Solution Manual for A First Course in Linear Algebra by Robert A. Beezer" 本书是罗伯特·A·比泽尔教授为《线性代数》编写的习题解答手册，版本号为3.00。比泽尔教授自1984年起在普吉特海湾大学担任数学教授，并在此期间进行教学与研究工作。他拥有圣克拉拉大学数学（计算机科学重点）学士学位、伊利诺伊大学香槟分校统计学硕士学位以及该校数学博士学位。比泽尔教授的教学足迹遍布全球，曾在西印度大学（特立尼达校区）和西澳大利亚大学进行学术访问，并在南非的非洲数学科学研究所的硕士课程中授课。自2008年起，他参与了开源数学软件Sage的开发工作，这表明他在数学教育和技术应用方面具有深厚的背景。比泽尔教授的常规教学课程包括微积分、线性代数和抽象代数，他的研究兴趣集中在将线性代数应用于图论。他的专业网站提供更多的教学和研究资料，网址为http://buzzard.ups.edu。该手册的最新版发布于2012年12月7日，封面设计未在摘要中详述。这本书的习题解答手册旨在辅助学生深入理解线性代数的基本概念，包括向量空间、线性映射、矩阵、特征值和特征向量、行列式、秩以及线性方程组的解等核心主题。通过解决书中的练习题，学生可以巩固课堂所学，提高解决问题的能力，为未来的数学和工程领域学习奠定坚实基础。比泽尔教授的经验和专业知识确保了这些习题和解答的质量，使得这本手册成为学习线性代数的宝贵资源。

12 Robert Beezer §RREF

So the unique solution is δ

= 2, δ

= −3, δ

= −2. Entirely similar work will lead you to

= −1 δ

= 1 δ

= 1

and

= −4 δ

= 8 δ

= 5

M45 (Chris Black) You keep a number of lizards, mice and peacocks as pets. There are a total of 108 legs

and 30 tails in your menagerie. You have twice as many mice as lizards. How many of each creature do you

have?

Solution (Chris Black) Let l, m, p denote the number of lizards, mice and peacocks. Then the statements

from the problem yield the equations:

4l + 4m + 2p = 108

l + m + p = 30

2l − m = 0

We form the augmented matrix for this system and row-reduce

4 4 2 108

1 1 1 30

2 −1 0 0

RREF

−−−−→





1 0 0 8

0 1 0 16

0 0 1 6





From the row-reduced matrix, we see that we have an equivalent system l = 8, m = 16, and p = 6, which

means that you have 8 lizards, 16 mice and 6 peacocks.

M50 (Robert Beezer) A parking lot has 66 vehicles (cars, trucks, motorcycles and bicycles) in it. There

are four times as many cars as trucks. The total number of tires (4 per car or truck, 2 per motorcycle or

bicycle) is 252. How many cars are there? How many bicycles?

Solution (Robert Beezer) Let c, t, m, b denote the number of cars, trucks, motorcycles, and bicycles. Then

the statements from the problem yield the equations:

c + t + m + b = 66

c − 4t = 0

4c + 4t + 2m + 2b = 252

We form the augmented matrix for this system and row-reduce

1 1 1 1 66

1 −4 0 0 0

4 4 2 2 252

RREF

−−−−→





1 0 0 0 48

0 1 0 0 12

0 0 1 1 6





The ﬁrst row of the matrix represents the equation c = 48, so there are 48 cars. The second row of the matrix

represents the equation t = 12, so there are 12 trucks. The third row of the matrix represents the equation

m + b = 6 so there are anywhere from 0 to 6 bicycles. We can also say that b is a free variable, but the

context of the problem limits it to 7 integer values since you cannot have a negative number of motorcycles.

T10 (Robert Beezer) Prove that each of the three row operations (Deﬁnition RO) is reversible. More

precisely, if the matrix B is obtained from A by application of a single row operation, show that there is a

single row operation that will transform B back into A.

Solution (Robert Beezer) If we can reverse each row operation individually, then we can reverse a sequence

of row operations. The operations that reverse each operation are listed below, using our shorthand notation.

Notice how requiring the scalar α to be non-zero makes the second operation reversible.

↔ R

αR

, α 6= 0

αR

+ R

− αR

+ R

T11 (Robert Beezer) Suppose that A, B and C are m ×n matrices. Use the deﬁnition of row-equivalence

(Deﬁnition REM) to prove the following three facts.

§TSS FCLA: Exercises and Solutions 13

1. A is row-equivalent to A.

2. If A is row-equivalent to B, then B is row-equivalent to A.

3. If A is row-equivalent to B, and B is row-equivalent to C, then A is row-equivalent to C.

A relationship that satisﬁes these three properties is known as an equivalence relation, an important

idea in the study of various algebras. This is a formal way of saying that a relationship behaves like equality,

without requiring the relationship to be as strict as equality itself. We’ll see it again in Theorem SER.

T12 (Robert Beezer) Suppose that B is an m ×n matrix in reduced row-echelon form. Build a new, likely

smaller, k × ` matrix C as follows. Keep any collection of k adjacent rows, k ≤ m. From these rows, keep

columns 1 through `, ` ≤ n. Prove that C is in reduced row-echelon form.

T13 (Robert Beezer) Generalize Exercise RREF.T12 by just keeping any k rows, and not requiring the

rows to be adjacent. Prove that any such matrix C is in reduced row-echelon form.

Section TSS

Types of Solution Sets

C10 (Robert Beezer) In the spirit of Example ISSI, describe the inﬁnite solution set for Archetype J.

For Exercises C21–C28, ﬁnd the solution set of the system of linear equations. Give the values of n and r,

and interpret your answers in light of the theorems of this section.

C21 (Chris Black)

+ 4x

+ 3x

− x

= 5

− x

+ x

+ 2x

= 6

+ x

+ 6x

+ 5x

= 9

Solution (Chris Black) The augmented matrix for the given linear system and its row-reduced form

are:

1 4 3 −1 5

1 −1 1 2 6

4 1 6 5 9

RREF

−−−−→





0 7/5 7/5 0

0 1 2/5 −3/5 0

0 0 0 0 1





For this system, we have n = 4 and r = 3. However, with a leading 1 in the last column we see that the

original system has no solution by Theorem RCLS.

C22 (Chris Black)

− 2x

+ x

− x

= 3

− 4x

+ x

= 2

− 2x

+ 3x

= 1

Solution (Chris Black) The augmented matrix for the given linear system and its row-reduced form

are:

1 −2 1 −1 3

2 −4 1 1 2

1 −2 −2 3 1

RREF

−−−−→





1 −2 0 0 3

0 0 1 0 −2

0 0 0 1 −2





Thus, we see we have an equivalent system for any scalar x

= 3 + 2x

= −2

16 Robert Beezer §TSS

M45 (Robert Beezer) The details for Archetype J include several sample solutions. Verify that one of

these solutions is correct (any one, but just one). Based only on this evidence, and especially without doing

any row operations, explain how you know this system of linear equations has inﬁnitely many solutions.

Solution (Robert Beezer) Demonstrate that the system is consistent by verifying any one of the four

sample solutions provided. Then because n = 9 > 6 = m, Theorem CMVEI gives us the conclusion that the

system has inﬁnitely many solutions. Notice that we only know the system will have at least 9 − 6 = 3 free

variables, but very well could have more. We do not know know that r = 6, only that r ≤ 6.

M46 (Manley Perkel) Consider Archetype J, and speciﬁcally the row-reduced version of the augmented

matrix of the system of equations, denoted as B here, and the values of r, D and F immediately following.

Determine the values of the entries

[B]

1,d

[B]

3,d

[B]

1,d

[B]

3,d

[B]

1,f

[B]

3,f

(See Exercise TSS.M70 for a generalization.)

For Exercises M51–M57 say as much as possible about each system’s solution set. Be sure to make it

clear which theorems you are using to reach your conclusions.

M51 (Robert Beezer) A consistent system of 8 equations in 6 variables.

Solution (Robert Beezer) Consistent means there is at least one solution (Deﬁnition CS). It will have

either a unique solution or inﬁnitely many solutions (Theorem PSSLS).

M52 (Robert Beezer) A consistent system of 6 equations in 8 variables.

Solution (Robert Beezer) With 6 rows in the augmented matrix, the row-reduced version will have

r ≤ 6. Since the system is consistent, apply Theorem CSRN to see that n −r ≥ 2 implies inﬁnitely many

solutions.

M53 (Robert Beezer) A system of 5 equations in 9 variables.

Solution (Robert Beezer) The system could be inconsistent. If it is consistent, then because it has

more variables than equations Theorem CMVEI implies that there would be inﬁnitely many solutions.

So, of all the possibilities in Theorem PSSLS, only the case of a unique solution can be ruled out.

M54 (Robert Beezer) A system with 12 equations in 35 variables.

Solution (Robert Beezer) The system could be inconsistent. If it is consistent, then Theorem CMVEI

tells us the solution set will be inﬁnite. So we can be certain that there is not a unique solution.

M56 (Robert Beezer) A system with 6 equations in 12 variables.

Solution (Robert Beezer) The system could be inconsistent. If it is consistent, and since 12 > 6, then

Theorem CMVEI says we will have inﬁnitely many solutions. So there are two possibilities. Theorem

PSSLS allows to state equivalently that a unique solution is an impossibility.

M57 (Robert Beezer) A system with 8 equations and 6 variables. The reduced row-echelon form of the

augmented matrix of the system has 7 pivot columns.

Solution (Robert Beezer) 7 pivot columns implies that there are r = 7 nonzero rows (so row 8 is all

zeros in the reduced row-echelon form). Then n + 1 = 6 + 1 = 7 = r and Theorem ISRN allows to

conclude that the system is inconsistent.

M60 (Robert Beezer) Without doing any computations, and without examining any solutions, say as much

as possible about the form of the solution set for each archetype that is a system of equations.

Archetype A, Archetype B, Archetype C, Archetype D, Archetype E, Archetype F, Archetype G, Archetype

H, Archetype I, Archetype J

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