**§ ****2 ****Linear spaces and linear subspaces**

1. Linear space

[ Linear operations ] Let *F* be a field whose elements *a* , *b* , *c* , ... as quantities; *V* be a set of objects of any kind whose elements are denoted by the Greek letters *α* , *β* , *γ* , ... . Determine the two algorithms : Addition of elements in 1

o ^{V. }*For* any two elements *α* and *β in **V* , there is always a unique definite element *γ* corresponding to them, which is called the sum of *α* and *β* , denoted as . 2 ^{o} The number in F is the same as that *in **V* Element-wise multiplication . For any number *a in **F* _{}

^{} * **With any element **α** in V* , there is always a unique and definite element *δ* corresponding to them, which is called the multiplication of *a* and *α , and is written as*_{}

These two algorithms are called linear operations .

[ Linear space and its properties ] Let *F* be a field and *V* be a set of objects of any kind. If the following conditions are satisfied for linear operations, then *V* is called a linear space on the field *F* : ( i) *V* is an additive group; ( ii) For any element *a* ∈ *F* and ** α** ∈

linear space over the real number field is called a real linear space; a linear space over the complex number field is called a complex linear space .

Example 1 All the vectors in the three-dimensional space form a real linear space .

Example 2 The polynomial ring *F* [ *x* ] on the number field *F* forms a linear space on the number field *F* according to the usual polynomial addition and polynomial multiplication .

Example 3 An *m* × *n* matrix whose elements belong to the number field *F* form a linear space on the number field *F* according to the addition of the matrix and the multiplication of the matrix and the number .

Example 4 According to usual addition and multiplication, the ensemble of real numbers is a linear space over the field **R** of real numbers . The ensemble of complex numbers is a linear space over the field C of complex numbers. Any field is a linear space **that** uses itself as the field of quantities .

Example 5 takes each continuous real function defined in a real interval ( *a* , *b* ) as an element, and the sum of any two elements *f* and g is denoted as a continuous real function defined in The value of a point *x* is specified as_{}_{}_{}

_{}

The element obtained by multiplying an element *f* by a real number *c* is defined as_{}

_{}

Then all these elements form a real linear space .

Linear spaces have the following properties:

1 ^{o} zero vector is unique .

2o ^{Negative} vectors are unique .

^{3o} . __{}

4 ^{o} If then *c* = 0 or ** α** =

[ Linearly dependent and linearly independent ] A finite set of vectors in a linear space *V* over the field *F* , if true , only then the equation_{}_{}_{}

_{}

The vector group is called linearly independent, otherwise it is called linearly dependent . If the vector group is linearly dependent, at least one of the vectors is a linear combination of the remaining vectors :_{}_{}_{}_{}

_{}

Any set of vectors containing the zero vector **0** is linearly dependent .

Assuming that convergence is defined on the linear space *V over the field **F* (Chapter 21, § 3 , 4) , a set of infinitely many vectors in *V* , A vector is said to be linearly independent if the time - only equation in *F* holds, otherwise it is said to be linearly dependent ._{}_{}_{}

_{}

_{}

[ Basis and coordinates ] A set of vectors in the linear space *V* on the field *F* if the_{}

(i) are linearly independent ;_{}

(ii) Any vector in *V* is a finite linear combination of vectors; then it is called a finite basis of *V* , also called generating (or expanding) this space, which is a set of generators of the space ._{}_{}_{}_{}

Set as a set of bases of *V* , then any vector *α** in V* must be represented by a linear combination:_{}_{}

_{}

The complex number in the formula is uniquely determined, and it is called the coordinate of the vector ** α** with respect to the base .

If *V* has a finite basis, it is called a finite-dimensional linear space, otherwise, it is called an infinite-dimensional space *.* The number of vectors of the basis of a finite-dimensional linear space *V* is called the dimension of V, denoted *by* ._{}

[ First Dimension Theorem ] Any two bases of a finite-dimensional linear space *V* over a field *F* have the same number of elements .

The inference is assumed to be a set of linearly independent vectors in an *n* -dimensional linear space *V.* Obviously , there is a basis in *V* that is a part of it . _{}_{}_{}

2. Linear subspace

[ Linear subspace ] Let *S* be a non-empty subset of the linear space *V* on the field *F. If the linear operation of **S* on *V* also constitutes a linear space, then *S* is called a linear subspace of *V* , which is referred to as a subspace for short .

Let *S* be a subset of the linear space *V* over the field *F , if closed with respect to linear operations, that is*

(i) if then ;_{}_{}

(ii) if , then ;_{}_{}

Then *S* is a subspace of *V.*

For example, a subset of a single zero vector in a linear space *V* is a subspace of *V* , called the zero subspace . *V* itself is also a subspace of
*V.* These two subspaces are called trivial subspaces of *V.*

Let be a set of vectors in the linear space *V* on the field *F* , all linear combinations of this set of vectors_{}

_{}_{}

A subspace that constitutes *V* is called the subspace generated (or stretched) by .
This is a non-trivial subspace of *V.*_{}

[ Intersection and Sum of Subspaces ] Let *S* , *T* be the subspace of the linear space *V* on the field *F* , and the subset formed by all the vectors in *V* belonging to *S* and *T* is called the intersection (communication set) of *S* and *T* , Denoted as . The subset composed of all vectors that can be represented as *S* and *T* is called the sum (sum set), denoted as (or ) ._{}_{}_{}_{}

Let *S* and *T* be two subspaces of a linear space *V on **F* , then the intersection and sum of *S* and *T* are both subspaces of *V.*_{}_{}

[ Second dimension theorem ] Let *S* and *T* be two subspaces of linear space *V* , then

_{}

(here denotes the dimension of the linear space *V* ) ._{}

It can be deduced that if the sum of the dimensions of the two subspaces *S* and *T in the **n* -dimensional linear space *V* is greater than *n* , then *S* and *T* must contain a common non-zero vector .

For example, two different planes (two-dimensional subspaces) in three-dimensional space intersect on a straight line, because , but , so ._{}_{}_{}

[ Direct sum of subspaces ] Let be a subspace of the linear space *V* , if the decomposition of each vector *α in the sum*_{}_{}

_{}

is unique . This sum is called a straight sum and is written as

_{}

Direct sums of subspaces have the following properties:

The necessary and sufficient conditions for ^{1o} to be a direct sum are:_{}

_{}

This is true only for all-zero vectors .
_{}

The necessary and sufficient conditions for the sum of ^{2o} to be a direct sum are:_{}

_{}*Φ* (empty set)_{}

3 ^{}Let ^{o} be a subspace of the linear space *V* , if_{}

_{}

but __ __ _{}

The reverse is also true .

This shows that for a direct sum of subspaces, the dimension is additive . It can be seen that if

_{}

the basis of the subspace_{}

_{}

Combined, we get a set of bases for the subspace *W.*

[ quotient space ] Let *S* be a subspace of *V* , and set two vectors , if , then the sum is equivalent, denoted as . In fact, this relation has three properties of equivalence relation: ( i) Reflexive For each , there are ; ( ii ) symmetry if , then ; ( iii) transitivity if , , then . As in the case of sets, two equivalent vector sums are said to belong to the same class .
Each vector contains exactly In a class, this class is denoted . The zero vector **0 in ***V* is contained in the class coincident with the subspace *S.*_{}_{}_{}_{}_{}

_{}_{}

_{}_{}

_{}_{}_{}

_{}_{}_{}_{}_{}

If each class is taken as an element, then the set of all these elements is a linear space, called the quotient space of *V* with respect to *S* , denoted as . The zero vector of the quotient space is , and has_{}_{}

_{}

It can be seen from this that if , the dimension of the quotient space is zero; and if *S* is a null space, the dimension of the quotient space is the same as that of *V.*_{}

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