线性代数几何入门第二版：Springer经典教材

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《线性代数通过几何：第二版》(Linear Algebra Through Geometry, 2nd edition) 是一本深入浅出的数学教材，由Springer出版社发行，适用于本科数学教育。本书作为"Undergraduate Texts in Mathematics"系列的一部分，旨在将线性代数的概念与几何直观相结合，以帮助读者更好地理解和掌握这一核心数学领域。该书由Axler、Gehring和Halmos三位编辑共同编撰，他们都是数学界知名的学者，他们的合作确保了内容的专业性和深度。作者们试图通过几何视角来解释抽象的线性代数理论，使得原本可能看似复杂的矩阵运算和向量空间理论变得生动易懂。书中可能会包含对向量、矩阵、线性变换、特征值与特征向量、秩和行列式等基本概念的详细讨论，以及如何运用这些概念解决几何问题的实例。例如，Banchoff和Wermer在书中可能引入了通过几何图形表示线性映射的方法，展示了如何用几何直观来理解线性方程组的解空间、几何变换的性质以及矩阵对几何形状的影响。此外，章节中可能还会探讨欧几里得几何、笛卡尔坐标系以及复数在几何上的应用。《线性代数通过几何：第二版》不仅是一本理论教材，还可能包含丰富的习题和练习，鼓励读者通过实践巩固所学知识。它可能适合那些希望从几何角度深化对线性代数理解的大学生、研究生以及教师，或是对几何与线性代数交叉领域感兴趣的专业人士。此外，书中可能还会参考其他数学史和哲学著作，如Anglin的《数学：简史与哲学》和Lambek的《泰勒斯遗产》，以及经典的数学教程如Apostol的《数论导引》、Armstrong的《基础拓扑学》和《群与对称性》等，为读者提供一个更广阔的数学文化背景。总体来说，《线性代数通过几何：第二版》是一本集理论严谨性和实践趣味性于一体的优秀教材，它将抽象的线性代数与直观的几何方法紧密结合起来，对于提升读者的数学素养和问题解决能力具有重要作用。

Linear Algebra Through Geometry

=(;)

=xE.

+yE.

xE.

Figure 2.3

Writing a vector in this way expresses the point as the fourth vertex

of a rectangle whose other three coordinates are

and (See

Fig. 2.3.)

More generally,

may obtain a geometric interpretation of vector

addition

follows.

start with the triangle with vertices

(

v+y

Figure

2.4

2.0 The Geometry of Vectors

the Plane

and move it parallel to itself so that its first vertex lies on then the

other two vertices lie on

and t

;),

respectively. (See Fig. 2.4.)

Thus, the sum of the vectors X

= (

and

U = ( can be obtained by

translating the directed segment from 0 to X parallel to itself until its

beginning point lies

The new endpoint

will

represent U +

and

this

will

be the fourth coordinate of a parallelogram with

and

X as the

other three vertices.

our

diagrams

have pictured addition of a vector X with positive

coordinates,

but

a similar argument shows that the parallelogram interpre-

tation is still valid if one or both coordinates are negative or zero.

referring either to the coordinate description or the geometric descrip-

tion,

can establish the following algebraic properties of vector addition

and scalar multiplication which are analogous to familiar facts about

arithmetic of numbers:

(4)

X+U=U+X.

Commutative law for vectors

(5)

+ U) + A = X + (U +

A).

Associative law for vectors

(6)

There

a vector 0 such that

X + 0 = X = 0 + X for all

Additive identity

(7)

For

any X there

a vector

- X such that X + ( -

Additive inverse

(8)

reX

+ U) =

+ rU.

Distributive law for vectors

(9)

+ s)(X) =

+ sX.

Distributive law for scalars

(10)

r(sX)

(rs)X.

Associative law for scalars

(II)

I . X = X for each

Note that it is possible for the parallelogram to collapse to a doubly

covered line segment if

add two multiples of the same vector. In Fig.

2.5,

show the parallelograms for B +

A +

and A + ( -

A).

We can use the negative of a vector to help define the notion of difference

U - X of the vectors X and

(See Fig. 2.6.) We define

U - X = U + ( -

X),

so, in coordinates,

(

- ( = U - X = U + ( -

= + ( =

;)

= =

;).

Since

(U-X)+X=U+«-X)+X)=U+O=U,

see that

U-X

the vector

add

to X to get

Thus,

move U - X parallel to itself

until its beginning point lies on

get the directed line segment from X

Thus,

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