Let two arbitrary Cartesian rectangular coordinate systems be given on the plane. The first is determined by the origin O and the basis vectors i j , the second - the center O' and basis vectors i ’ j ’ .
Let's set the goal to express the coordinates x y of some point M with respect to the first coordinate system through x’ and y’ are the coordinates of the same point relative to the second system.
notice, that
Let's denote the coordinates of the point O' with respect to the first system as a and b:
Let's decompose the vectors i ’ and j ’ basis i j :
(*)
In addition, we have: 
. We introduce here expansions of vectors in terms of the basis i
’
j
’
:
from here 
It can be concluded that no matter what two arbitrary Cartesian systems on the plane are, the coordinates of any point of the plane with respect to the first system are linear functions of the coordinates of the same point with respect to the second system.
We multiply the equations (*) scalarly first by i , then on j :
O 
denote by the angle between the vectors i
and i
’
. Coordinate system i
j
can be combined with the system i
’
j
’
by parallel translation and subsequent rotation by an angle . But here an arc option is also possible: the angle between the basis vectors i
i
’
also , and the angle between the basis vectors j
’
j
’
equals - . These systems cannot be combined with parallel translation and rotation. You also need to change the direction of the axis at to the opposite.
From the formula (**) we obtain in the first case:
In the second case
The conversion formulas are:
We will not consider the second case. Let us agree that both systems are right.
Those. conclusion: whatever the two right-handed coordinate systems, the first of them can be combined with the second by parallel translation and subsequent rotation around the origin by some angle .
Parallel transfer formulas: 
Axis rotation formulas: 
Reverse transformations:
Transformation of Cartesian rectangular coordinates in space.
In space, reasoning in a similar way, we can write:
(***)
And for coordinates get:
(****)
So, whatever two arbitrary coordinate systems in space, the x y z coordinates of some point with respect to the first system are linear functions of the coordinates x’ y’ z’ the same point with respect to the second coordinate system.
Multiplying each of the equalities (***) scalarly by i ’ j ’ k ’ we get:
V 
Let us clarify the geometric meaning of the transformation formulas (****). To do this, assume that both systems have a common origin: a
=
b
=
c
= 0
.
Let us introduce into consideration three angles that completely characterize the location of the axes of the second system relative to the first.
The first angle is formed by the x-axis and the u-axis, which is the intersection of the xOy and x’Oy’ planes. The direction of the angle is the shortest turn from the x to y axis. Let's denote the angle as . The second angle is the angle between the axes Oz and Oz’ not exceeding . Finally, the third angle is the angle between the u-axis and Ox', measured from the u-axis in the direction of the shortest turn from Ox' to Oy'. These angles are called Euler angles.
The transformation of the first system into the second can be represented as a succession of three rotations: through the angle relative to the Oz axis; at an angle relative to the Ox’ axis; and at an angle relative to the Oz’ axis.
Numbers ij can be expressed in terms of Euler angles. We will not write these formulas because of their cumbersomeness.
The transformation itself is a superposition of parallel translation and three successive Euler angle rotations.
All these considerations can also be carried out for the case when both systems are left, or have different orientations.
If we have two arbitrary systems, then, generally speaking, they can be combined by parallel translation and one rotation in space around some axis. We will not look for her.
1) Transition from one Cartesian rectangular coordinate system in the plane to another Cartesian rectangular system with the same orientation and the same origin.
Assume that two Cartesian rectangular coordinate systems are introduced on the plane hoy and with a common origin O having the same orientation (Fig. 145). Let us denote the unit vectors of the axes Oh and OU respectively through and , and the unit vectors of the axes and through and . Finally let - the angle from the axis Oh up to the axis. Let X and at– coordinates of an arbitrary point M in system hoy, and and are the coordinates of the same point M in system .
Since the angle from the axis Oh up to the vector is , then the coordinates of the vector
Angle off axis Oh up to the vector is ; so the coordinates of the vector are .
Formulas (3) § 97 take the form
Transition matrix from one Cartesian hoy rectangular coordinate system to another rectangular coordinate system with the same orientation is
A matrix is called orthogonal if the sum of the squares of the elements located in each column is equal to 1, and the sum of the products of the corresponding elements of different columns is equal to zero, i.e. if
Thus, the transition matrix (2) from one rectangular coordinate system to another rectangular system with the same orientation is orthogonal. Note also that the determinant of this matrix is +1:
Conversely, if an orthogonal matrix (3) with a determinant equal to +1 is given, and a Cartesian rectangular coordinate system is introduced on the plane hoy, then, by virtue of relations (4), the vectors and are unit and mutually perpendicular, therefore, the coordinates of the vector in the system hoy are equal to and , where is the angle from the vector to the vector , and since the vector is unit and we get from the vector by turning to , then either , or .
The second possibility is ruled out, since if we had , then we are given that .
So, and the matrix A has the form
those. is the transition matrix from one rectangular coordinate system hoy to another rectangular system having the same orientation, with angle .
2. Transition from one Cartesian rectangular coordinate system on the plane to another Cartesian rectangular system with the opposite orientation and with the same origin.
Let two Cartesian rectangular coordinate systems be introduced on the plane hoy and with a common origin O, but having the opposite orientation, denote the angle from the axis Oh to the axis through (the orientation of the plane will be set by the system hoy).
Since the angle from the axis Oh up to the vector is , then the coordinates of the vector are:
Now the angle from vector to vector is (Fig. 146), so the angle from the axis Oh up to the vector is equal (according to the Chall theorem for angles) and therefore the coordinates of the vector are equal:
And formulas (3) § 97 take the form
Transition matrix
orthogonal, but its determinant is -1. (7)
Conversely, any orthogonal matrix with determinant equal to -1 specifies the transformation of one rectangular coordinate system on the plane into another rectangular system with the same origin but opposite orientation. So, if two Cartesian coordinate systems hoy and have a common beginning, then
where X, at– coordinates of any point in the system hoy; and are the coordinates of the same point in the system , and
orthogonal matrix.
Conversely, if
arbitrary orthogonal matrix, then by the relations
expresses the transformation of a Cartesian rectangular coordinate system into a Cartesian rectangular system with the same origin; - coordinates in the system hoy a unit vector giving the positive axis direction ; - coordinates in the system hoy a unit vector giving the positive axis direction.
coordinate systems hoy and have the same orientation, and in the case - the opposite.
3. General transformation of one Cartesian rectangular coordinate system on the plane into another rectangular system.
Based on points 1) and 2) of this paragraph, as well as on the basis of § 96, we conclude that if rectangular coordinate systems are introduced on the plane hoy and , then the coordinates X and at arbitrary point M planes in the system hoy with coordinates of the same point M in the system are related by relations - the coordinates of the origin of the coordinate system in the system hoy.
Note that the old and new coordinates X, at and , the vectors under the general transformation of the Cartesian rectangular coordinate system are related by the relations
if the systems hoy and have the same orientation and relations
if these systems have the opposite orientation, or in the form
orthogonal matrix. Transformations (10) and (11) are called orthogonal.
Let two arbitrary Cartesian rectangular coordinate systems be given on the plane. The first is determined by the origin O and the basis vectors i j , the second - the center O' and basis vectors i ’ j ’ .
Let's set the goal to express the coordinates x y of some point M with respect to the first coordinate system through x’ and y’ are the coordinates of the same point relative to the second system.
notice, that
Let's denote the coordinates of the point O' with respect to the first system as a and b:
Let's decompose the vectors i ’ and j ’ basis i j :
(*)
In addition, we have: 
. We introduce here expansions of vectors in terms of the basis i
’
j
’
:
from here 
It can be concluded that no matter what two arbitrary Cartesian systems on the plane are, the coordinates of any point of the plane with respect to the first system are linear functions of the coordinates of the same point with respect to the second system.
We multiply the equations (*) scalarly first by i , then on j :
Denote by the angle between the vectors i and i ’ . Coordinate system i j can be combined with the system i ’ j ’ by parallel translation and subsequent rotation by an angle . But here an arc option is also possible: the angle between the basis vectors i i ’ also , and the angle between the basis vectors j ’ j ’ equals - . These systems cannot be combined with parallel translation and rotation. You also need to change the direction of the axis at to the opposite.
From the formula (**) we obtain in the first case:
In the second case
The conversion formulas are:
We will not consider the second case. Let us agree that both systems are right.
Those. conclusion: whatever the two right-handed coordinate systems, the first of them can be combined with the second by parallel translation and subsequent rotation around the origin by some angle .
Parallel transfer formulas: 
Axis rotation formulas: 
Reverse transformations:
Transformation of Cartesian rectangular coordinates in space.
In space, reasoning in a similar way, we can write:
(***)
And for coordinates get:
(****)
So, whatever two arbitrary coordinate systems in space, the x y z coordinates of some point with respect to the first system are linear functions of the coordinates x’ y’ z’ the same point with respect to the second coordinate system.
Multiplying each of the equalities (***) scalarly by i ’ j ’ k ’ we get:
V 
Let us clarify the geometric meaning of the transformation formulas (****). To do this, assume that both systems have a common origin: a
=
b
=
c
= 0
.
Let us introduce into consideration three angles that completely characterize the location of the axes of the second system relative to the first.
The first angle is formed by the x-axis and the u-axis, which is the intersection of the xOy and x’Oy’ planes. The direction of the angle is the shortest turn from the x to y axis. Let's denote the angle as . The second angle is the angle between the axes Oz and Oz’ not exceeding . Finally, the third angle is the angle between the u-axis and Ox', measured from the u-axis in the direction of the shortest turn from Ox' to Oy'. These angles are called Euler angles.
The transformation of the first system into the second can be represented as a succession of three rotations: through the angle relative to the Oz axis; at an angle relative to the Ox’ axis; and at an angle relative to the Oz’ axis.
Numbers ij can be expressed in terms of Euler angles. We will not write these formulas because of their cumbersomeness.
The transformation itself is a superposition of parallel translation and three successive Euler angle rotations.
All these considerations can also be carried out for the case when both systems are left, or have different orientations.
If we have two arbitrary systems, then, generally speaking, they can be combined by parallel translation and one rotation in space around some axis. We will not look for her.
Topic 5. Linear transformations.
Coordinate systemcalled a method that allows using numbers to uniquely establish the position of a point relative to some geometric figure. Examples are the coordinate system on a straight line - the coordinate axis and rectangular Cartesian coordinate systems, respectively, on the plane and in space.
Let's perform the transition from one coordinate system xy on the plane to another system , i.e. Let us find out how the Cartesian coordinates of the same point in these two systems are related.
Consider first parallel transfer rectangular Cartesian coordinate system xy, i.e. the case when the axes and of the new system are parallel to the corresponding axes x and y of the old system and have the same directions with them.
If the coordinates of the points M (x; y) and (a; b) in the xy system are known, then (Fig. 15) in the system the point M has the coordinates: .
Let the segment OM of length ρ form an angle with the axis and . Then (Fig. 16) with the x axis, the segment OM forms an angle and the coordinates of the point M in the xy system are 
, 
.
Taking into account that in the coordinate system of the point M are equal and , we obtain
When turning through the angle "clockwise", respectively, we get: 
Task 0.54. Determine the coordinates of the point M(-3; 7) in the new coordinate system x / y / , the origin 0 / of which is at the point (3; -4), and the axes are parallel to the axes of the old coordinate system and are equally directed with them.
Solution. We substitute the known coordinates of the points M and O / into the formulas: x / = x-a, y / = y-b.
We get: x / = -3-3=-6, y / = 7-(-4)=11. Answer: M / (-6; 11).
§2. The concept of a linear transformation, its matrix.
If each element x of the set X, according to some rule f, corresponds to one and only one element y of the set Y, then we say that display f of the set X into the set Y, and the set X is called domain of definition mappings f . If, in particular, the element x 0 н X corresponds to the element y 0 н Y, then we write y 0 = f (х 0). In this case, the element at 0 is called way element x 0 , and element x 0 - prototype element at 0 . The subset Y 0 of the set Y, consisting of all images, is called set of values mappings f.
If, under a mapping f, different elements of the set X correspond to different elements of the set Y, then the mapping f is called reversible.
If Y 0 \u003d Y, then the mapping f is called the mapping of the set X on the set Y.
An invertible mapping of a set X onto a set Y is called one-to-one.
Particular cases of the concept of mapping a set to a set are the concept numeric function and concept geometric display.
If the mapping f maps to each element of the set X a unique element of the same set X, then such a mapping is called transformation sets of X.
Let the set of n-dimensional vectors of the linear space L n be given.
The transformation f of an n-dimensional linear space L n is called linear transformation if
for any vectors from L n and any real numbers α and β. In other words, a transformation is called linear if a linear combination of vectors goes into a linear combination of their images with the same coefficients.
If a vector is given in some basis and the transformation f is linear, then by definition , where are the images of the basis vectors.
Therefore, the linear transformation is completely is defined, if the images of the basis vectors of the considered linear space are given:
(12)
Matrix 
in which the kth column is the coordinate column of the vector 
in basis, called matrix linear transformations f in this basis.
The determinant det L is called the determinant of the transformation f and Rg L is called the rank of the linear transformation f.
If the matrix of a linear transformation is non-degenerate, then the transformation itself is also non-degenerate. It transforms one-to-one the space L n into itself, i.e. each vector from L n is the image of some unique vector of it.
If the matrix of a linear transformation is degenerate, then the transformation itself is degenerate. It transforms the linear space L n into some part of it.
Theorem.As a result of applying the linear transformation f with the matrix L to the vector 
the result is a vector 
such that .
The numbers written in brackets are the coordinates of the vector according to the basis:
|
By definition of the operation of matrix multiplication, system (13) can be replaced by the matrix
equality 
, which was to be proved.
Exampleslinear transformations.
1. Stretching along the x-axis by k 1 times, and along the y-axis by k 2 times on the xy plane is determined by the matrix and the coordinate transformation formulas are: x / = k 1 x; y / = k 2 y.
2. Mirror reflection about the y-axis on the xy plane is determined by the matrix and the coordinate transformation formulas are: x / = -x, y / = y.