线性代数学习指南：错误与反思

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"linear algebra done wrong" 《线性代数做错了》是一本由Sergei Treil编写的数学教材，特别针对线性代数的基础知识，对于编程人员来说是必修课程。这本书在国外享有较高的知名度，被列为必读书目。作者在Brown University的数学系工作，并且对教材内容拥有版权，其遵循的是Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License，允许非商业用途的自由使用，包括学习和教学。读者可以免费获取电子版，也可以打印纸质副本供个人或教学使用。本书的前言中，作者提出了书名“线性代数做错了”的疑问。尽管书名听起来有些反常，但作者的意图并非指出错误的教学方法，而是采用了一种独特的方式教授线性代数。这本书最初是作为“荣誉线性代数”课程的讲义而编写的，目标读者是对线性代数有深入理解需求的学生，特别是那些首次接触该主题的人。书中可能将传统上被认为是“正确”的线性代数方法与现代的、更直观的方法进行对比，通过这种方式激发学生对这个基础数学领域的兴趣和理解。例如，作者可能会探讨如何通过矩阵的几何解释来教授向量空间、线性映射和特征值，而不是仅仅停留在抽象的定义和公式上。这种“做错”的方式实际上是一种创新的教学策略，旨在帮助学生更好地掌握概念，而不是死记硬背公式。线性代数是计算机科学、工程、物理等许多领域的基石，它包括向量、矩阵、行列式、特征值和特征向量、线性变换、秩和null空间等核心概念。在编程中，理解这些概念对于处理数据、建模问题和设计算法至关重要。通过《线性代数做错了》这本教材，读者不仅能够学习到线性代数的基本理论，还能了解到如何将这些理论应用于实际问题解决中。作者可能强调了线性代数的几何直觉和应用，这对于编程人员尤其重要，因为编程经常涉及到二维和三维空间的数据操作。例如，线性代数在图形学中的应用，如变换坐标、光照计算以及动画制作，都需要对向量和矩阵有深刻的理解。此外，机器学习和数据科学领域也大量使用线性代数，如矩阵分解用于推荐系统，特征提取和降维等。《线性代数做错了》是一本试图打破传统教学模式，通过新颖视角教授线性代数的教材。它鼓励读者从不同的角度理解和应用线性代数，这对于提升编程技能和解决实际问题的能力具有深远的影响。

6 1. Basic Notions

2. Linear combinations, bases.

Let V be a vector space, and let v

, v

, . . . , v

∈ V be a collection of vectors.

A linear combination of vectors v

, v

, . . . , v

is a sum of form

+ α

+ . . . + α

k=1

Deﬁnition 2.1. A system of vectors v

, v

, . . . v

∈ V is called a basis (for

the vector space V ) if any vector v ∈ V admits a unique representation as

a linear combination

v = α

+ α

+ . . . + α

k=1

The coeﬃcients α

, α

, . . . , α

are called coordinates of the vector v (in the

basis, or with respect to the basis v

, v

, . . . , v

Another way to say that v

, v

, . . . , v

is a basis is to say that for any

possible choice of the right side v, the equation x

+. . .+x

= v

(with unknowns x

) has a unique solution.

Before discussing any properties of bases

, let us give a few examples,

showing that such objects exist, and that it makes sense to study them.

Example 2.2. In the ﬁrst example the space V is F

, where F is either R

or C. Consider vectors













, e













, e













, . . . , e













(the vector e

has all entries 0 except the entry number k, which is 1). The

system of vectors e

, e

, . . . , e

is a basis in F

. Indeed, any vector

v =













∈ F

can be represented as the linear combination

v = x

+ x

+ . . . x

k=1

the plural for the “basis” is bases, the same as the plural for “base”

2. Linear combinations, bases. 7

and this representation is unique. The system e

, e

, . . . , e

∈ F

is called

the standard basis in F

Example 2.3. In this example the space is the space P

of the polynomials

of degree at most n. Consider vectors (polynomials) e

, e

, . . . , e

∈ P

deﬁned by

:= 1, e

:= t, e

:= t

, e

:= t

, . . . , e

:= t

Clearly, any polynomial p, p(t) = a

t+a

+. . .+a

admits a unique

representation

p = a

+ a

+ . . . + a

So the system e

, e

, . . . , e

∈ P

is a basis in P

. We will call it the

standard basis in P

Remark 2.4. If a vector space V has a basis v

, v

, . . . , v

, then any vector

v is uniquely deﬁned by its coeﬃcients in the decomposition v =

k=1

. This is a very im-

portant remark, that

will be used through-

out the book. It al-

lows us to translate

any statement about

the standard column

space F

to a vector

space V with a basis

, v

, . . . , v

So, if we stack the coeﬃcients α

in a column, we can operate with them

as if they were column vectors, i.e. as with elements of F

(again here F is

either R or C, but everything also works for an abstract ﬁeld F).

Namely, if v =

k=1

and w =

k=1

, then

v + w =

k=1

(α

+ β

i.e. to get the column of coordinates of the sum one just need to add the

columns of coordinates of the summands. Similarly, to get the coordinates

of αv we need simply to multiply the column of coordinates of v by α.

2.1. Generating and linearly independent systems. The deﬁnition

of a basis says that any vector admits a unique representation as a linear

combination. This statement is in fact two statements, namely that the rep-

resentation exists and that it is unique. Let us analyze these two statements

separately.

If we only consider the existence we get the following notion

Deﬁnition 2.5. A system of vectors v

, v

, . . . , v

∈ V is called a generating

system (also a spanning system, or a complete system) in V if any vector

v ∈ V admits representation as a linear combination

v = α

+ α

+ . . . + α

k=1

The only diﬀerence from the deﬁnition of a basis is that we do not assume

that the representation above is unique.

8 1. Basic Notions

The words generating, spanning and complete here are synonyms. I per-

sonally prefer the term complete, because of my operator theory background.

Generating and spanning are more often used in linear algebra textbooks.

Clearly, any basis is a generating (complete) system. Also, if we have a

basis, say v

, v

, . . . , v

, and we add to it several vectors, say v

n+1

, . . . , v

then the new system will be a generating (complete) system. Indeed, we can

represent any vector as a linear combination of the vectors v

, v

, . . . , v

and just ignore the new ones (by putting corresponding coeﬃcients α

= 0).

Now, let us turn our attention to the uniqueness. We do not want to

worry about existence, so let us consider the zero vector 0, which always

admits a representation as a linear combination.

Deﬁnition. A linear combination α

+ α

+ . . . + α

is called trivial

if α

= 0 ∀k.

A trivial linear combination is always (for all choices of vectors

, v

, . . . , v

) equal to 0, and that is probably the reason for the name.

Deﬁnition. A system of vectors v

, v

, . . . , v

∈ V is called linearly inde-

pendent if only the trivial linear combination (

k=1

with α

= 0 ∀k)

of vectors v

, v

, . . . , v

equals 0.

In other words, the system v

, v

, . . . , v

is linearly independent iﬀ the

equation x

+ x

+ . . . + x

= 0 (with unknowns x

) has only trivial

solution x

= x

= . . . = x

= 0.

If a system is not linearly independent, it is called linearly dependent.

By negating the deﬁnition of linear independence, we get the following

Deﬁnition. A system of vectors v

, v

, . . . , v

is called linearly dependent

if 0 can be represented as a nontrivial linear combination, 0 =

k=1

Non-trivial here means that at least one of the coeﬃcient α

is non-zero.

This can be (and usually is) written as

k=1

|α

| 6= 0.

So, restating the deﬁnition we can say, that a system is linearly depen-

dent if and only if there exist scalars α

, α

, . . . , α

k=1

|α

| 6= 0 such

that

k=1

= 0.

An alternative deﬁnition (in terms of equations) is that a system v

, . . . , v

is linearly dependent iﬀ the equation

+ x

+ . . . + x

= 0

(with unknowns x

) has a non-trivial solution. Non-trivial, once again

means that at least one of x

is diﬀerent from 0, and it can be written

k=1

| 6= 0.

10 1. Basic Notions

Proof. We already know that a basis is always linearly independent and

complete, so in one direction the proposition is already proved.

Let us prove the other direction. Suppose a system v

, v

, . . . , v

is lin-

early independent and complete. Take an arbitrary vector v ∈ V . Since the

system v

, v

, . . . , v

is linearly complete (generating), v can be represented

v = α

+ α

+ . . . + α

k=1

We only need to show that this representation is unique.

Suppose v admits another representation

v =

k=1

eα

Then

k=1

(α

− eα

k=1

−

k=1

eα

= v − v = 0.

Since the system is linearly independent, α

− eα

= 0 ∀k, and thus the

representation v = α

+ α

+ . . . + α

is unique. 

Remark. In many textbooks a basis is deﬁned as a complete and linearly

independent system (by Proposition 2.7 this deﬁnition is equivalent to ours).

Although this deﬁnition is more common than one presented in this text, I

prefer the latter. It emphasizes the main property of a basis, namely that

any vector admits a unique representation as a linear combination.

Proposition 2.8. Any (ﬁnite) generating system contains a basis.

Proof. Suppose v

, v

, . . . , v

∈ V is a generating (complete) set. If it is

linearly independent, it is a basis, and we are done.

Suppose it is not linearly independent, i.e. it is linearly dependent. Then

there exists a vector v

which can be represented as a linear combination of

the vectors v

, j 6= k.

Since v

can be represented as a linear combination of vectors v

, j 6= k,

any linear combination of vectors v

, v

, . . . , v

can be represented as a linear

combination of the same vectors without v

(i.e. the vectors v

, 1 ≤ j ≤ p,

j 6= k). So, if we delete the vector v

, the new system will still be a complete

one.

If the new system is linearly independent, we are done. If not, we repeat

the procedure.

Repeating this procedure ﬁnitely many times we arrive to a linearly

independent and complete system, because otherwise we delete all vectors

and end up with an empty set.

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