M. Thamban Nair与Arindama Singh合作的线性代数权威指南

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《M. Thamban Nair & Arindama Singh 线性代数》是一本于2018年出版的专业数学教材，由印度理工学院马德拉斯分校的M. Thamban Nair和Arindama Singh合著。本书是针对线性代数领域的深入研究，内容涵盖了线性代数的核心概念和技术。第1章“向量空间”介绍了基本的向量理论，包括向量的定义、性质以及向量空间的构造，这为后续章节奠定了坚实的基础。读者可以在此了解向量的加法、标量乘法以及向量空间的封闭性和线性组合等概念。第2章“线性变换”讨论了函数在向量空间中的行为，包括线性映射、矩阵表示以及线性变换与矩阵之间的关系。这部分内容对于理解矩阵运算和线性系统的重要性不言而喻。第3章“基本操作”涉及对向量空间中的元素进行操作，如求解线性方程组、特征向量和特征值的初步探讨。这些操作在工程、科学和计算机科学中广泛应用，例如在数据处理和机器学习中。第4章“内积空间”扩展了向量的结构，引入了内积的概念，如欧几里得空间中的点积，这对于计算长度、角度和正交性至关重要。内积空间还涉及范数和完备性，这些概念在信号处理和优化问题中发挥关键作用。第5章“特征值和特征向量”是线性代数的核心内容之一，通过分析矩阵的行为，研究如何找到那些使得矩阵与其相乘只改变标量倍数的特殊向量，即特征向量和对应的数值，即特征值。这一章不仅有助于理解矩阵的对角化，还对动力学系统和谱理论有深远影响。第6章“块对角化表示”探讨了将复杂数学模型简化的方法，通过将矩阵分解为独立部分，便于处理和分析。这种技术在大型系统的简化处理和高效算法设计中非常实用。第7章“谱表示”进一步深化了对矩阵特性的理解，特别是通过谱理论，研究线性算子的本征值和本征函数，这对量子力学、信号处理和统计力学等领域具有重要意义。该书提供了丰富的理论背景和实例，适合高等教育机构的学生和研究人员，以及对线性代数有深入需求的专业人士。书中附带的ISBN号码以及电子书版本链接，表明它是一本可供全球范围内的读者获取的权威参考文献。版权声明强调了所有权利保留，确保了学术诚信和尊重知识产权。

1.1 Vector Space 5

or as A =[a

] for short, with a

∈ F for i = 1,...,m; j = 1,...,n. The number

which occurs at the entry in i th row and j th column is referred to as the (i, j )th

entry of the matrix [a

]. For A =[a

] and B =[b

] in F

m×n

, and α ∈ F, we deﬁne

A + B =[a

+ b

]∈F

m×n

,αA =[αa

]∈F

m×n

We say that two matrices are equal when their respective entries are equal. That is,

for A =[a

] and B =[b

], we write A = B if and only if a

= b

. With these

operations of addition and scalar multiplication, F

m×n

becomes a vector space over

F. The zero vector in F

m×n

is the zero matrix, i.e. the matrix with all entries 0, and

the additive inverse of A =[a

]∈F

m×n

is the matrix −A := [−a

(3) For n ∈{0, 1, 2,...},letP

(F) denote the set of all polynomials (in the variable

t)ofdegreeatmostn, with coefﬁcients in F. That is, x ∈ P

(F) if and only if x is

of the form

x = a

+ a

t +···+a

for some scalars a

, a

,...,a

. Here, we assume that a scalar is a polynomial of

degree 0. Further, two polynomials are considered equal when the coefﬁcients of

respective powers of t are equal. That is,

+ a

t +···+a

= b

+ b

t +···+b

if and only if a

= b

for i = 0,...,n.

Addition and scalar multiplication on P

(F) are deﬁned as follows. For

x = a

+ a

t +···+a

, y = b

+ b

t +···+b

in P

(F), and α ∈ F,

x + y := (a

+ b

) + (a

+ b

)t +···+(a

+ b

αx := αa

+ αa

t +···+αa

The zero vector in P

(F) is the polynomial with all its coefﬁcients zero, and

−(a

+ a

t +···+a

) =−a

− a

t −···−a

Then P

(F) is a vector space.

(4) Let P(F) =∪

∞

n=0

(F), the set of all polynomials with coefﬁcients in F. That

is, x ∈ P(F) if and only if x = a

+ a

t +···+a

for some n ∈{0, 1, 2,...} and

for some scalars a

,...,a

∈ F.

If x, y ∈ P (F), then x ∈ P

(F) and y ∈ P

(F) for some m, n. So, x , y ∈ P

(F),

with k = max {m, n}. The equality relation, addition, and scalar multiplication are

deﬁned as in (3). Then P(F) is a vector space.

(5) Let V be the set of all s equences of scalars. A sequence whose nth term is a

written as (a

). Two s equences are considered equal if and only if their respective

terms are equal, that is, (a

) = (b

) if and only if a

= b

for each n ∈ N. For

), (b

) ∈ V , and α ∈ F, deﬁne

6 1 Vector Spaces

) + (b

) := (a

+ b

), α(a

) := (αa

That is,

, a

,...)+ (b

, b

,...)= (a

+ b

, a

+ b

, a

+ b

,...),

α(a

, a

....)= (αa

,αa

,...).

With this addition and scalar multiplication, V is a vector space, where its zero vector

is the sequence with each term as zero, and −(a

) = (−a

). This space is called the

sequence space and is denoted by F

∞

(6) Let S be a nonempty set. Let V be a vector space over F. Let F (S, V ) be the set of

all functions from S into V . As usual, x = y for x, y ∈ F (S, V ) when x(s) = y(s)

for each s ∈ S. For x, y ∈ F (S, V ) and α ∈ F, deﬁne x + y and αx point-wise; that

is,

(x + y)(s) := x(s) + y(s), (αx)(s) := αx(s)

for s ∈ S.

Let the functions 0 and −x in F (S, V ) be deﬁned by

0(s) = 0,(−x)(s) =−x(s) for s ∈ S.

Then F (S, V ) is a vector space over F with the zero vector as 0 and the additive

inverse of x as −x. We sometimes refer to this space as a function space. 

Comments on Notation: P

(R) denotes the real vector space of all polynomials of

degree at most n with real coefﬁcients. P

all polynomials of degree at most n with complex coefﬁcients. Similarly, P(R) is the

real vector space of all polynomials with real coefﬁcients, and P(C) is the complex

vector space of all polynomials with complex coefﬁcients. Note that C is also a vector

space over R. Similarly, P

if V is a complex vector space, then it is also a real vector space. If at all we require

to regard any vector space over C also as a vector space over R, we will speciﬁcally

mention it.

As particular cases of Example 1.3(2), (Read: Example 1.3 Part 2) we have the

vector spaces F

m×1

,thesetofallcolumn vectors of size m, and F

1×n

,thesetof

all row vectors of size n. To save space, we use the transpose notation in writing

a column vector. That is, a column vector v of size n with its entries a

,...,a

written as

⎡

⎢

⎣

⎤

⎥

⎦

or as [a

··· a

]

By putting the superscript T over a row vector v we mean that the column vector is

obtained by taking transpose of the row vector v. When the column vectors are writ-

ten by the lower case letters u, v, w, x, y, z with or without subscripts (sometimes

1.1 Vector Space 7

superscripts), the corresponding row vectors will be written with the transpose nota-

tion, that is, as u

, v

. Further, we will not distinguish between the square brackets

and the parentheses. Usually, we will write a row vector with parentheses. Thus, we

will not distinguish between

··· a

] and (a

,...,a

Thus we regard F

same as F

1×n

. We may recall that by taking transpose of a matrix

in F

m×n

, we obtain a matrix in F

n×m

.Thatis,ifA =[a

]∈F

m×n

, then

:= [b

]∈F

n×m

with b

= a

for i = 1,...,m; j = 1,...,n.

Many vector spaces can be viewed as function spaces. For example, with S = N

and V = F, we obtain the sequence space of Example 1.3(5). With S ={1,...,n}

and V = F, each function in F(S, F) can be speciﬁed by an n-tuple of its function

values. Therefore, the vector space F ({1,...,n}, F) can be viewed as F

and also

as F

n×1

. Some more examples of function spaces follow.

Example 1.4 (1) Let I be an interval, and let C(I, R) denote the set of all real-valued

continuous functions deﬁned on I .Forx, y ∈ C(I, R) and α ∈ R, deﬁne x + y and

αx point-wise as in Example 1.3(6).

The functions x + y and αx are in C(I, R). Then C(I, R) is a real vector space

with the zero element as the zero function and the additive inverse of x ∈ C(I, R) as

the function −x deﬁned by (−x)(t) =−x (t ) for all t ∈ I.

(2) Let R([a, b],

R) denote the set of all real-valued Riemann integrable functions

on [a, b]. Deﬁne addition and scalar multiplication point-wise, as in Example 1.3(6).

From the theory of Riemann integration, it follows that if x, y ∈ R([a, b], R) and

α ∈ R, then x + y,αx ∈ R([a, b], R). It is a real vector space.

(3) For k ∈ N,letC

([a, b], F) denote the set of all functions x from [a, b] to F such

that the kth derivative x

(k)

exists and is continuous on [a, b].

Deﬁne addition and scalar multiplication point-wise, as in Example 1.3(6). Then

([a, b], F) is a vector space. Notice that

([a, b], R) ⊆ C([a, b ], R) ⊆ R([a, b], R) ⊆ F ([a, b], R). 

Example 1.5 Let V

,...,V

be vector spaces over F. Consider the Cartesian product

V = V

×···×V

={(x

,...,x

) : x

∈ V

,...,x

∈ V

Deﬁne addition and scalar multiplication on V by

,...,x

) + (y

,...,y

) := (x

+ y

,...,x

+ y

α(x

,...,x

) := (αx

,...,αx

1.1 Vector Space 9

Theorem 1.6 allows us to do algebraic manipulation of vectors like scalars as long

as addition is concerned. Thus we abbreviate x + (−y) to x − y. However, there is

a big difference: vectors cannot be multiplied or raised to powers, in general. Look

at the proof of Theorem 1.6(6). We used α

−1

instead of x

−1

It follows from Theorem 1.6(7) that every nontrivial vector space contains inﬁnitely

many vectors.

Exercises for Sect. 1.1

In the following a set V , a ﬁeld F, which is either R or C, and the operations of

addition and scalar multiplication are given. Check whether V is a vector space over

F, with these operations.

1. With addition and scalar multiplication as i n R

(a) V ={(a, b) ∈ R

: 2a + 3b = 0}, F = R.

(b) V ={(a, b) ∈ R

: a + b = 1}, F = R.

: ab = 0}, F = R.

2. V = R

, F = R, with addition as in R

, and scalar multiplication as given by the

following: for (a, b) ∈ V,α∈ R,

(a) α(a, b ) := (a, 0).

(b) α(a, b ) := (b,αa).

(d) α(a, b ) :=



(0, 0) if α = 0

(αa, b/α) if α = 0.

3. V ={x ∈ R : x > 0}, F = R, and for x, y ∈ V,α

∈ R, x + y := xy,αx :=

. The operations on the left are deﬁned by the known operations on the right.

4. V ={x ∈ R : x ≥ 0}, F = R, and for x, y ∈ V,α∈ R, x + y := xy,αx :=

|α|x.

5. V = C

, F = C, and for x = (a, b), y = (c, d), α ∈ C,

x + y := (a + 2c, b + 3d), αx := (αa,αb).

6. V is the set of all polynomials of degree 5 with real coefﬁcients, F = R, and the

operations are the addition and scalar multiplication of polynomials.

7. S is a nonempty set, s ∈ S, V is the set of all functions f : S → R with f (s) = 0,

F = R, and the operations are the addition and scalar multiplication of functions.

8. V is the set of all functions f : R → C satisfying f (−t) =

f (t), F = R, and the

operations are the addition and scalar multiplication of functions.

9. V ={x}, where x is some symbol, and addition and scalar multiplication are

deﬁned as x + x = x,αx = x for all α ∈ F.

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