# Preface

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# The articles

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （1）： Second and third order determinants 、 Total permutation and its reverse order number

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （2）：n Step determinant 、 exchange

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （3）： The nature of determinants

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （4）： Determinant by line （ Column ） an

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （5）： Kramer's law

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （6）： Matrix operation

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （7）： Inverse matrix

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （8）： Elementary transformation of matrix

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （9）： The rank of a matrix 、 Solutions of linear equations

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （10）： Vector group and its linear combination

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （11）： Linear correlation of vector groups

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （12）： The rank of a vector group

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （13）： Structure of solutions of linear equations

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （14）： Vector space

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （15）： The inner product of a vector 、 Length and orthogonality

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （16）： Eigenvalues and eigenvectors of square matrices

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （17）： Similarity matrix

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （18）： Diagonalization of symmetric matrices

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （19）： Quadratic form and its canonical form

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （20）： The quadratic form is transformed into standard form by formula method

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （21）： Positive definite quadratic form

# 6.1 Definition and properties of linear space

## Definition 1： Linear space

set up V V It's a Nonempty set , R \mathbb{R} by Real number field

If for any two elements α , β ∈ V \alpha,\beta \in V , There is always one element γ ∈ V \gamma \in V With the corresponding , be called α \alpha And β \beta And , Write it down as γ = α + β \gamma=\alpha+ \beta

For any number λ ∈ R , α ∈ V \lambda\in \mathbb{R},\alpha \in V , There is always one element δ ∈ V \delta\in V With the corresponding , be called λ \lambda And α \alpha Product of , Write it down as δ = λ α \delta=\lambda \alpha

And these two kinds of transportation meet Eight operation rules

1. α + β = β + α \alpha+\beta=\beta+\alpha
2. ( α + β ) + γ = α + ( β + γ ) (\alpha+\beta)+\gamma=\alpha+(\beta+\gamma)
3. stay V V There is a zero element in 0 \boldsymbol0 , To any α ∈ V \alpha\in V , There are α + 0 = α \alpha+\boldsymbol0=\alpha
4. To any α ∈ V \alpha \in V , There are α \alpha The negative element of β ∈ V \beta \in V , send α + β = 0 \alpha+\beta=\boldsymbol0
5. 1 α = α 1\alpha=\alpha
6. λ ( μ α ) = ( λ μ ) α \lambda(\mu\alpha)=(\lambda \mu)\alpha
7. ( λ + μ ) α = λ α + μ α (\lambda+\mu)\alpha=\lambda \alpha+\mu\alpha
8. λ ( α + β ) = λ α + λ β \lambda(\alpha+\beta)=\lambda \alpha+\lambda\beta

notes ： α , β , γ ∈ V ; λ , u ∈ R \alpha,\beta,\gamma \in V;\lambda,u\in \mathbb{R}

that V V It's called the real number field R \mathbb{R} Upper Vector space （ Or linear space ）

in short

• All addition and multiplication that satisfy the above eight laws operation , It's called Linear operation
• Any set that defines a linear operation , It's called vector space

## Properties of linear spaces

### nature 1

The zero element is unique

prove （ Reduction to absurdity ）

Suppose there are two zero elements 0 1 , 0 2 ∈ V 0_1,0_2 \in V

According to the definition of zero element , Yes

{ 0 1 + 0 2 = 0 1 ( 0 2 see become zero element plain ) 0 1 + 0 2 = 0 2 ( 0 1 see become zero element plain ) \begin{cases} 0_1 + 0_2=0_1(0_2 As a zero element )\\ 0_1 + 0_2=0_2(0_1 As a zero element )\\ \end{cases}

obtain

0 1 = 0 2 0_1=0_2

It shows that the zero element is unique

### nature 2

The negative element of any element is unique , α \alpha The negative element of is written as − α -\alpha

prove （ Reduction to absurdity ）

hypothesis α ∈ V \alpha \in V There are two negative elements , Write it down as β , γ \beta,\gamma

According to the definition of negative element , Yes

{ α + β = 0 α + γ = 0 \begin{cases} \alpha + \beta = 0\\ \alpha + \gamma = 0 \end{cases}

also

β = β + 0 = β + ( α + γ ) = ( β + α ) + γ = 0 + γ = γ \beta=\beta+0=\beta+(\alpha+\gamma)=(\beta+\alpha)+\gamma=0+\gamma=\gamma

namely

β = γ \beta=\gamma

Sum up , The negative element of any element is unique

### nature 3

（ 1 ） 0 α = 0 （1）0\alpha=\boldsymbol0
（ 2 ） ( − 1 ) α = − α （2）(-1)\alpha=-\alpha
（ 3 ） λ 0 = 0 （3）\lambda \boldsymbol0=\boldsymbol0

Prove （1）

α + 0 α = 1 α + 0 α = ( 1 + 0 ) α = α \alpha+0\alpha=1\alpha+0\alpha=(1+0)\alpha=\alpha

obtain

0 α = 0 0\alpha=\boldsymbol0

Prove （2）

α + ( − 1 ) α = ( 1 − 1 ) α = 0 \alpha+(-1)\alpha=(1-1)\alpha=\boldsymbol0

According to the definition of negative element , obtain

( − 1 ) α = − α (-1)\alpha=-\alpha

Prove （3）

λ 0 = λ ( α + ( − 1 ) α ) = λ α + ( − λ ) α = ( λ + ( − λ ) ) α = 0 α = 0 \lambda \boldsymbol0=\lambda(\alpha+(-1)\alpha)=\lambda\alpha+(-\lambda)\alpha=(\lambda+(-\lambda))\alpha=0\alpha=\boldsymbol0

namely

λ 0 = 0 \lambda \boldsymbol0=\boldsymbol0

### nature 4

If λ α = 0 \lambda \alpha=\boldsymbol0 , be λ = 0 \lambda=0 or α = 0 \alpha=\boldsymbol0

prove

When λ = 0 \lambda=0 when , λ α = 0 \lambda \alpha=\boldsymbol0

When λ ≠ 0 \lambda\neq0 when ,

equation λ α = 0 \lambda \alpha=\boldsymbol0 Ride on both sides 1 λ \frac{1}{\lambda} , have to

1 λ ( λ α ) = 1 λ 0 = 0 \frac{1}{\lambda}(\lambda \alpha)=\frac{1}{\lambda}\boldsymbol0=\boldsymbol0

Again because

1 λ ( λ α ) = ( 1 λ λ ) α = 1 α = α \frac{1}{\lambda}(\lambda \alpha)=(\frac{1}{\lambda}\lambda)\alpha=1\alpha=\alpha

Introduction

α = 0 \alpha=\boldsymbol0

（2） The necessity of proof ：

from λ = 0 \lambda=0 or α = 0 \alpha=\boldsymbol0

It's easy to get λ α = 0 \lambda \alpha=\boldsymbol0

Sum up , If λ α = 0 \lambda \alpha=\boldsymbol0 , be λ = 0 \lambda=0 or α = 0 \alpha=\boldsymbol0

## give an example

### example 1

explain P [ x ] n P[x]_n It's vector space , among P [ x ] n P[x]_n Indicates that the number of times does not exceed n All of the polynomials of

P [ x ] n = { a n x n + a n − 1 x n − 1 + . . . + a 1 x + a 0 ∣ a n , . . . , a 1 , a 0 ∈ R } P[x]_n=\{a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0|a_n,...,a_1,a_0 \in \mathbb{R} \}

prove

Prove the closure of addition operation ：

set up

α ∈ P [ x ] n , β ∈ P [ x ] n \alpha\in P[x]_n,\beta\in P[x]_n

Yes

α + β ∈ P [ x ] n \alpha + \beta \in P[x]_n

α + β \alpha + \beta The number of times of any item in the will not exceed n n , So the result also belongs to P [ x ] n P[x]_n

The closure of the multiplication of the number of certificates ：

set up

k ∈ R , α ∈ P [ x ] n k\in \mathbb{R},\alpha \in P[x]_n

Yes

k α ∈ P [ x ] n k\alpha \in P[x]_n

Multiplication does not make P [ x ] n P[x]_n The maximum number of times exceeded n, The result is also in P [ x ] n P[x]_n in

The addition of polynomials 、 The number multiplication operation satisfies the law of linear operation , That is, eight operation laws , No more details here

Sum up , P [ x ] n P[x]_n It's vector space

### example 2

explain Q [ x ] n Q[x]_n It's vector space , among Q [ x ] n Q[x]_n Expressed as

Q [ x ] n = { a n x n + a n − 1 x n − 1 + . . . + a 1 x + a 0 ∣ a n , . . . , a 1 , a 0 ∈ R , And a n ≠ 0 } Q[x]_n=\{a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0|a_n,...,a_1,a_0 \in \mathbb{R}, And a_n\neq0 \}

prove

Prove the closure of addition operation ：

set up

α ∈ Q [ x ] n , β ∈ Q [ x ] n \alpha \in Q[x]_n,\beta \in Q[x]_n

Yes

α + β ∈ Q [ x ] n \alpha + \beta \in Q[x]_n

The closure of the multiplication of the number of certificates ：

set up

k ∈ R , α ∈ Q [ x ] n k\in \mathbb{R},\alpha \in Q[x]_n

obtain k α k\alpha It doesn't necessarily belong to Q [ x ] n Q[x]_n

The special case is when k = 0 k=0 when

k α = 0 α = 0 k\alpha=0\alpha=0

Be careful Q [ x ] n Q[x]_n In the definition a n ≠ 0 a_n\neq0 , explain Q [ x ] n Q[x]_n It must be non-zero

So number multiplication is not closed

Sum up , Q [ x ] n Q[x]_n It's not vector space

### example 5

The totality of positive real numbers , Write it down as R + \mathbb{R}^+ , In which the addition and multiplication operations are defined as

{ a ⊕ b = a b ( a , b ∈ R + ) λ ⊙ a = a λ ( λ ∈ R , a ∈ R + ) \begin{cases} a\oplus b=ab(a,b\in \mathbb{R}^+)\\ \lambda \odot a=a^{\lambda}(\lambda\in \mathbb{R},a\in \mathbb{R^+} ) \end{cases}

Test certificate R + \mathbb{R}^+ The above addition and multiplication operations form a linear space

prove

Prove the closure of addition operation ：

To any a , b ∈ R + a,b\in\mathbb{R}^+ , There are

a ⊕ b = a b ∈ R + a\oplus b=ab \in \mathbb{R}^+

The closure of the multiplication of the number of certificates ：

To any λ ∈ R , a ∈ R + \lambda\in \mathbb{R},a\in \mathbb{R^+} , Yes

λ ⊙ a = a λ ∈ R + \lambda \odot a=a^{\lambda}\in \mathbb{R^+}

Prove eight operation laws ：

It's not proved one by one here , The result is that all eight operation laws meet

But it should be noted that

At this time The zero element is 1

That is, for any a ∈ R + a \in \mathbb{R^+} , Yes a ⊕ 1 = a a\oplus1=a

Just follow the negative element 、 Definition of zero element , Then we can solve it according to our custom operation rules

### Summary

（1） Prove whether a set constitutes a vector space , Definitely not only To verify the addition 、 Closure of number multiplication operation

（2） When the defined addition and multiplication operations are not the usual addition and multiplication operations between real numbers , It also needs to prove whether eight linear operation laws are satisfied

（3） When proving uniqueness , have access to Reduction to absurdity , Suppose there are multiple elements at the same time , Then prove that these elements are equal .

# Conclusion

explain ：

• Refer to textbook 《 linear algebra 》 The fifth edition Department of mathematics, Tongji University
• Cooperate with the concept explanation in the book Combined with some of my own understanding and thinking

The article is only for study notes , Record from 0 To 1 A process of

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