Matrix Theory 3 linear transformation and its matrix

Third, linear transformation and its matrix

I. linear transformation and its operations

Definition: SetVYesfieldKLinear Space on,TYesVA ing to itself enablesVAny element inXAll have uniqueYVCorresponding to it, it is calledTIsVOneTransformOrOperator,

Tx = y

NameYIsXIn the transformationTBottom image,XIsY.

IfTMeet

T (kx + ly) = k (Tx) + l (Ty)
X, yV, k, lK

NameTIsLinear transformation.

[Example 1] the operation of turning a vector space around the Rotation Angle of the origin is a linear transformation.

[Proof]

This operation is visible.TIt is a linear transformation.

,K, l

TLinear transformation.

[Example 2] All real polynomials whose number of times does not exceed constitute a linear space of a dimension in the real number field. The basis is optional, and a differential operator is a linear transformation on.

[Proof] It is obviously a transformation,

It must be proved that the conditions for linear transformation are met.

,K, l

Is a linear transformation.

2. Nature

1. Linear transformation changes zero elements to zero elements
   
2. The negative element is the negative element of the original element.
   
3. Linear transformation changes linear correlation element groups into linear correlation element groups.
   

[Proof] linear transformationT (kx + ly) = k (Tx) + l (Ty)

(1)T (0) = T (0x) = 0 (Tx) =0

(2)T(-X) = (-1)(Tx) =-(Tx)

(3) linear correlation of element groups, that is, there is a set of incomplete zero numbers

Then

Linear correlation.

[Proof]

It should be noted that linear independent element groups are not necessarily Linearly Independent after linear transformation. The transformed element groups are related to linear transformation. If linear transformation still transforms all element groups into linear independent element groups, it is called a full-rank linear transformation. The transformation matrix is a full-rank matrix.

3. linear transformation operation

1. Constant Transformation:
2. Zero Transformation:
3. The transformation is equal: Yes, two linear transformations, both, are called =
4. Sum of linear transformations + :,
5. Multiplication of linear transformations :,

   Negative Transformation:

6. The product of linear transformation :,
7. Inverse Transformation: If linear transformation exists, it is called inverse transformation =
8. Polynomial of linear transformation:
   

And specify

Note:

1) also known as the unit transformation, its matrix is represented as the unit matrix;

2) the corresponding matrix is represented as a zero matrix;

3) Like the product of a matrix, the product of linear transformation does not meet the exchange law;

4) Not all transformations have inverse transformations. Only full rank transformations have inverse transformations ,;

5) constant transformation, zero transformation, sum of linear transformation, product polynomial, and inverse transformation (if any) are linear transformation.

 

2. Matrix Representation of linear transformation

Linear Transformations are represented by matrices, which convert abstract linear transformations into specific matrix forms.

It is a linear transformation of linear space, and is a base, n, with a unique coordinate representation.

=

Therefore, to determine the linear transformation, you only need to determine the image of the base element in the change.

For any element, the transformed coordinates are represented

At the same time

Comparison:

=

That is:

1. Definition: it is called a matrix under the base.
   
2. Theorem: Set a base, and the matrices under this base are ,. Then there is
   

   (1)

   (2)

   (3)

   (4)

   Inference 1. Set to pure volumeTOfMPolynomial, a linear transformation of linear space, and the matrix under the base is
   

   
   

   Where

   
   

   Inference 2. Let the matrix of the linear transformation under the base be, and the coordinate of the element under the base is, then the coordinates under the base satisfy
   

   
   =

3. similarity Matrix

The two bases and matrices are respectively sum, and =, then

That is, it is a similarity matrix.

[Proof]

That is

Theorem: The necessary and sufficient conditions for the first square matrix and similarity are matrices in different bases for the same linear transformation.

[Proof] necessity: known and similar, I .e. there is a reversible Matrix

Select a base and define

Can be used as the basis and

Returns the matrix of the same linear transformation under different bases.

The proof of adequacy is provided by the proof of the definition of the similarity matrix.

 

3. linear transformation and the value and core of the matrix

1. Definition: linear transformation of a linear space, called
   

Value range;

It is called the core.

And are all subspaces.

Set as level matrix, called

Is the value range of the matrix;

For the core.

, Called the rank and zero degree;

, Called the rank and zero degree.

1. Theorem: (1)

(2)

(3) indicates the number of columns.

If it is a matrix of linear transformation, then

=, =

 

 

Job: P77-78, 1, 26, 7