**§ ****3 ****Linear Transformation**

1. Basic Concepts

[ Linear transformation ] Let the sum be two linear spaces on the same field *F* , and the mappings satisfy the following two conditions:_{}_{}_{}

(i) _{}, for any ;_{}

(ii) _{}, for any ;_{}

** Then L** is called a linear mapping or a linear transformation, also known as a homomorphism . If and are the same linear space, then

Example 1 A linear function on a linear space *V* (see Section 3) is a linear transformation of *V* to a field *F* (considered as a one-dimensional linear space) ._{}

Example 2 is assumed to be a linear function on a linear space *V , then by*_{}

_{}

The determined mapping is a linear transformation from *V* to *m* -dimensional space ._{}

Example 3 Let *V* be a real linear space composed of all continuous functions on the interval [ *a* , *b* ] . If let

_{}

Then ** L** is a linear transformation of

_{}

Example 4 Let *V* be a linear space composed of all real coefficient polynomials *f* ( *x* ) . If let

_{} ( the derivative of )_{}_{}

Then ** L** is a linear transformation of

[ Properties of Linear Transformations ]

1 ^{o} Conditions (i) , (ii) in the definition of linear transformation are equivalent to: for any_{}

_{}

Applying this formula repeatedly, export

_{}

2 If ^{o} is linearly independent, it is a linear transformation, then_{}_{}

_{}

is also linearly independent .

3 ^{o} If it constitutes a basis of *V* , and let , then there is a unique linear transformation ** L** such that .

[ Zero Transformation·Identity Transformation·Inverse Transformation ] The transformation that transforms any vector ** α** in the linear space

_{} ( the zero vector of )_{}_{}

Transforming any vector ** α** in the linear space

_{}

Both the zero transformation and the identity transformation are linear transformations .

On the linear transformation ** L** , if there is a linear transformation

[ Matrix of linear transformation ] Let it be a set of bases of the linear space *V* , and the base is a linear transformation, then it can be expressed as_{}_{}_{}_{}_{}

_{}

matrix of coefficients

_{}

is called a matrix of linear transformation ** L** with respect to bases { }

In particular, when *V* is the same dimension as V, or ** L** is a linear transformation of

After the basis is determined, the linear transformation and its matrix establish a one-to-one correspondence . The matrix of the zero transformation is the zero matrix, and the matrix of the identity transformation is the identity matrix .

[ Eigenvalues and eigenvectors of linear transformation ] , if present , such that the automorphism satisfies_{}_{}

_{}

Then the eigenvalues (eigenroots) of the linear transformation ** L** are called eigenvectors corresponding to .

The eigenvalues and eigenvectors of a linear transformation are equal to the eigenvalues and eigenvectors of the transformed matrix, respectively .

[ Rank of image , image source , kernel , linear transformation ] If it is a linear transformation, it is called the image of V, and V is called *the **image* source, and it is called the kernel . The dimension is called the rank of ** L** , and the dimension is called Degeneration times .

The kernel and image of a linear transformation are linear subspaces of *V* and V , respectively, and the sum of the dimensions of the kernel and the image is equal to the dimension of the image source . That is_{}_{}_{}_{}

_{}

The rank of a linear transformation is equal to the rank of the transformed matrix .

Second, the operation of linear transformation

[ Sum and multiplication of linear transformations ] The set of linear transformations from space *V* to space , denoted as_{}

_{}

is defined according to the following formula :_{}_{}

_{}

Both new transformations are linear, and

_{}

_{}They are called the sum and multiplication of linear transformations, respectively .

By the sum and number multiplication of the linear transformations defined above, the set forms a linear space on F. *Its* dimension is equal to the product of the dimensions *n* and *m* of the sum of *V.*_{}_{}_{}

[ The product of linear transformation ] is set to three linear spaces, if , then define_{}_{}_{}

_{} _{}

Obviously a linear transformation from , called the product of linear transformations ._{}_{}_{}

The product of linear transformations satisfies:

1 ^{o} distributive law if then _{}

_{}

2 ^{o} Associative law if . _{}

_{}

[ Idempotent transformation ] If ** L** is a linear transformation of a linear space

_{}

Then ** L** is called an idempotent transformation .

[ Isomorphism and Automorphism ] If the linear transformation is one-to-one, then ** L** is called isomorphism, or

Isomorphism has the following properties:

The necessary and sufficient conditions for ^{1o} to be an isomorphism are:_{}

_{}

2 ^{o} If ** L** and

_{}

In particular, for automorphisms , the above formula also holds ._{}

3 The set ^{}*G* formed by all the automorphisms of the linear space *V* on the field *F* forms a group under multiplication . *G* is called the linear transformation group of V, denoted by , *where* n ^{is }*the* dimension of *V.*_{}

4. The set ^{}*R* formed by all the linear transformations (automorphisms) of the linear space *V on the *^{o} field *F* forms a ring under addition and multiplication, and *R* is called the linear transformation ring of *A.*

3. Dual space and dual mapping

[ Quantity product and dual space ] Let the sum of *V* be two real (complex) linear spaces .
If a quantity is determined for any pair of vectors , and the following conditions are satisfied:_{}_{}_{}

(i)
_{}

_{}

(ii) For a fixed and all , if then ; conversely, for a fixed and all , if then . The function is called quantity product ._{}_{}_{}_{}_{}_{}_{}_{}_{}

If , then it is said to be orthogonal . (ii) shows that a vector in one space is orthogonal to all vectors in another space, only if it is a zero vector ._{}_{}

Two linear spaces that define the product of quantities are called dual spaces .

Dual spaces have equal dimensions .

[ Dual basis ] If the two basis sums of *V* and V satisfy the relation:_{}_{}_{}

_{}

Then they are called dual bases .

*V* and V are dual spaces, then for a known base of *V* , there is exactly one dual base ._{}_{}_{}_{}

[ Orthogonal Complementary Space ] Let it be a subspace of V, then the set of vectors that are orthogonal to all vectors in space *V* is a subspace of *V **,* called the orthogonal complementary space, denoted as ._{}_{}_{}_{}_{}_{}_{}

Orthogonal complementary spaces have the following properties:

1 ^{o} The sum of the dimensions of the space sum is equal to the dimension of the space *V* , that is_{}_{}

_{}

^{2o} _^{}_{}

3 ^{o} If , then ; and the sum is a pair of dual spaces, and the sum is also a pair of dual spaces ._{}_{}_{}_{}_{}_{}

[ Conjugate space ] Let *V* be a linear space on the field *F* , if yes , there is a unique number corresponding to F on F *,* then this correspondence is called a function defined on *V.*_{}_{}_{}_{}

function

_{}

For any two vectors and any , we have_{}_{}

_{}_{}

Then it is called a linear function, also known as a linear functional .
Let , then there is , so it is also called a linear homogeneous function or linear type ._{}_{}_{}

* *The sum and multiplication of two functions of the set of linear functions in *V are defined in the usual way as follows:*_{}_{}_{}

_{}

Then a linear space is formed, called the conjugate space of *V* , where the zero vector is a function that is always equal to zero .
_{}_{}_{}

It can be shown that and *V* are a pair of dual spaces, if { } is a set of basis of *V* , then the function defined by the following equation is a basis of:_{}_{}_{}_{}

_{}

Thus { } _{}is again the conjugate base of { } _{}.

[ Dual mapping ] Let *V* , and *W* , be two pairs of dual spaces; if two linear mappings:_{}_{}

_{}and_{}

For everything and everything , there is_{}_{}

_{}

** Then L** is called a dual mapping .

Dual mappings have the following properties:

1 ^{O} For a known linear map , there is exactly one dual map ._{}_{}

The rank of the ^{2O} dual map ** L** sum is equal .

3 ^{O} A sufficient and necessary condition for a vector to be contained in the image space is that it is orthogonal to all vectors in the kernel .
_{}_{}_{}_{}

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