Table of contents

Linear algebra studies linear mappings on finite-dimensional vector spaces .

1.AR^n and C^n

complex number :

A complex number is an ordered pair (a, b), where $a,b\in R$ , is generally written as $a+bi$ , where $i^2=-1$
The set of all complex numbers is denoted C: $C=\{a+bi:a,b\in R\}$
Addition and multiplication in C are defined as

$(a+bi)+(c+di)=(a+c)+(b+d)i,$

$(a+bi)(c+di)=(ac-bd)+(ad+bc)i,$

in $a,b,c,d\in R.$

Arithmetic properties of complex numbers :

Commutativity, $\alpha,\beta\in C$ available to all $\alpha+\beta=\beta+\alpha,\alpha \beta=\beta\alpha$ ;
Associativity, for $\alpha,\beta\in C$ all $(\alpha+\beta)+\lambda=\alpha+(\beta+\lambda),(\alpha\beta)\lambda=\alpha(\beta\lambda)$ ;
identities, for $\lambda\in C$ all $\lambda +0=\lambda,\lambda1=\lambda$
Additive inverse (additive inverse), $\alpha\in C$ there is a unique $\beta \in C$ such that for each $\alpha+\beta=0$ ;
Multiplicative inverse (multiplicative inverse), $\alpha\in C,\alpha \ne0$ there is a unique $\beta\in C$ such that for each $\alpha\beta=1$ ;
Distributive property, $\lambda,\alpha,\beta \in C$ available to all $\lambda(\alpha+\beta)=\lambda\alpha+\lambda\beta$

Group (list), length (length) :

Let n be a non-negative integer. A group of length n is a sequence of n elements separated by commas and enclosed in parentheses (these elements can be numbers, other groups, or something more abstract). A group of length n has the form:

$(x_1,...,x_n)$

Two groups are equal if and only if they are of equal length, contain the same elements, and are in the same order.

F : F represents R (set of real numbers) or C (set of complex numbers) . The letter F was chosen because both R and C are examples of fields.

F^n : $F^n$ It is a set of groups of length n composed of elements in F.

$F^n=\{(x_1,...,x_n):x_j\in F,j=1,...,n\}$

For $(x_1,...,x_n)\in F^n$ and $j\in \{1,...,n\}$ , it is called $x_j$ the th coordinate $(x_1,...,x_n)$ of $j$ .

Addition in F^n $F^n$ :

$F^n$ The addition in is defined as the addition of corresponding coordinates,

$(x_1,...,x_n)+(y_1,...,y_n)=(x_1+y_1,...,x_n+y_n)$

Additive commutativity of F^n :

If $x,y\in F^n$ , then $x+y=y+x$ .

0 : Use 0 to represent a group of length n and all coordinates are 0, $0=(0,...,0).$

Additive inverse in F^n :

For $x\in F^n$ , the additive inverse of x (denoted as -x) is a vector that satisfies the following conditions $-x\in F^n$ :

$x+(-x)=0.$

In other words, if $x=(x_1,...,x_n)$ , then $-x=(-x_1,...,-x_n)$ .

Scalar multiplication in F^n :

The product of a number λ and $F^n$ a vector in is calculated by multiplying each coordinate of the vector by λ, that is

$\lambda(x_1,...,x_n)=(\lambda x_1,...,\lambda x_n),$

in $\lambda \in F,(x_1,...,x_n)\in F^n.$

1.B Definition of vector space

Addition, scalar multiplication :

Addition on a set V is a function that associates each pair with $u,v\in V$ an $V$ element $u+v$ .
$V$ Scalar multiplication on a set is a function that maps $\lambda\in F$ any sum $v\in V$ to an element $\lambda v\in V$ .

Vector space :

A vector space is a set V with addition and scalar multiplication , satisfying the following properties

Commutativity, $u,v\in V$ available to all $u+v=v+u$ ;
Associativity, there is a sum for all $u,v,w\in V$ sums ; $a,b\in F$ $(u+v)+w=u+(v+w)$ $(ab)v=a(bv)$
Additive identity, elements exist such that they exist $0\in V$ for all ; $v\in V$ $v+0=v$
Additive inverse $v\in V$ exists for each such $w\in V$ that $v+w=0$ ;
Multiplicative identity, $v\in V$ available for all $1v=v$ ;
Distributive properties (distributive properties), there is a sum for all $a,b\in F$ sums . $u,v\in V$ $a(u+v)=au+av$ $(a+b)v=av+bv$

Vector, point : Elements in vector space are called vectors or points.

Real vector space (real vector space), complex vector space (complex vector space) :

A vector space on R is called a real vector space.
The vector space on C is called a complex vector space.

Notation F^S :

Let $S$ be a set, let represent the set of all functions from $F^S$ S. $F$
For $f,g\in F^S$ , it is stipulated that $f+g\in F^S$ the sum is the following function: for all $x\in S$ ,

$(f+g)(x)=f(x)+g(x)$ .

For $\lambda\in F$ sums $f\in F^S$ , the product $\lambda f\in F^S$ is stipulated to be the following function: for all $x\in S$ ,

$(\lambda f)(x)=\lambda f(x)$ .

The additive identity element is unique : The vector space has a unique additive identity element.

The additive inverse is unique : Each element in the vector space has a unique additive inverse.

1.C subspace

Subspace : If a subset U of V (using the same addition and scalar multiplication as V) is also a vector space, then U is said to be a subspace of V.

Conditions of subspace : A subset U of V is a subspace of V if and only if U satisfies the following three conditions:

Additive identity, $0\in U$ ;
closed under addition, $u,w\in U$ implication $u+w \in U$ ;
Closed under scalar multiplication, $a\in F$ and $u\in U$ implication $in the U$ .

Sum of subsets :

Assume $U_1,...,U_m$ that all $V$ are subsets of , then the sum $U_1,...,U_m$ of is defined as the set of all possible $U_1,...,U_m$ sums of the elements in , denoted as . more specifically, $U_1,...,U_m$

$U_1,...,U_m=\{u_1+...+u_m:u_1\in U_1,...,u_m\in U_m\}$ .

The sum of subspaces is the smallest subspace that contains these subspaces : assuming $U_1,...,U_m$ all are $V$ subspaces of , then $U_1,...,U_m$ isthe smallest subspace $V$ that contains $U_1,...,U_m$

Direct sum :

Assume $U_1,...,U_m$ that all $V$ are subspaces,

A sum is called a direct sum if $U_1+...+U_m$ each element in the sum can be uniquely expressed as $u_1+...+u_m$ , where each element $u_j$ belongs to . $U_j$ $U_1+...+U_m$
If the sum $U_1+...+U_m$ is a direct sum, it is $U_1\oplus...\oplus+U_m$ used to indicate $U_1+...+U_m$ that the symbol here $\oplus$ indicates that the sum here is a direct sum.

Conditions for direct sum :

Let $U_1,...,U_m$ all be $V$ subspaces of . It is a direct sum if and only if "each is equal to 0, which is the only way $u_j$ to express 0 as ". $u_1+...+u_n$ $U_1+...+U_m$

Direct sum of two subspaces :

Suppose $U$ and $W$ are $V$ both subspaces, if and only $U\cap W=\{0\}$ then , $U+W$ they are direct sums.

[Linear algebra should be learned this way] Chapter 1 Vector Space

1.AR^n and C^n

1.B Definition of vector space

1.C subspace

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