Linear and affine transformations of the plane

if $\mathcal{A}_{1} \subseteq \mathcal{A}_{2} \subseteq \mathcal{X}$, then $f\left(\mathcal{A}_{1}\right) \subseteq f\left(\mathcal{A}_{2}\right)$

$f\left(\mathcal{A}_{1} \cup \mathcal{A}_{2}\right) = f\left(\mathcal{A}_{1}\right) \cup f\left(\mathcal{A}_{2}\right)$

if $\mathcal{B}_{1}, \mathcal{B}_{2} \subseteq f(\mathcal{X}) \subseteq \mathcal{Y}$, then $f^{ - 1}\left(\mathcal{B}_{1} \cap \mathcal{B}_{2}\right) = f^{ - 1}\left(\mathcal{B}_{1}\right) \cap f^{ - 1}\left(\mathcal{B}_{2}\right)$

$f\left(\mathcal{A}_{1} \cap \mathcal{A}_{2}\right) \subseteq f\left(\mathcal{A}_{1}\right) \cap f\left(\mathcal{A}_{2}\right)$. Can the image of an intersection fail to coincide with the intersection of the images?

How many transformations exist of a set consisting of $n$ elements? How many of them are one-to-one?

How many possibilities are there for the sets of values of transformations of a set of $n$ elements?

How many mappings exist of a set of $m$ elements into a set of $n$ elements?

A surjection when $n > m$?

An injection when $n < m$?

$f$ is an injection

$f$ is a surjection

$f$ is one-to-one.

$f$ is a surjection, but not an injection

$f$ is an injection, but not one-to-one.

the set of all integers

the set of all even numbers

the set of all rational numbers

the set of all points of the plane whose coordinates are rational (rational points)

the set of all intervals on the line with rational endpoints

the set of all disks in the plane with centers at rational points and rational radii

the set of all polynomials $p(x) = a_{0} + a_{1} x + \ldots + a_{n} x^{n} (n = 0, 1, 2, \ldots )$ with integer coefficients $a_{i} \quad (i = 0, 1, \ldots , n)$.

no one-to-one correspondence can be established between the set of all integers and the set of all sequences of the numbers 0 and 1

there exists a one-to-one correspondence between the set of all real numbers and the set of all sequences of the numbers 0 and 1.

$\mathcal{Z}$ is finite

$\mathcal{Z}$ is countable.

onto the interval $(- 1, 1)$

onto the segment $\left[ - 1, 1\right]$.

onto the open disk $x^{2} + y^{2} < 1$

onto the closed disk $x^{2} + y^{2} \leqslant 1$

onto the square $\left|x\right| < 1, \left|y\right| < 1$ (the coordinate system is rectangular).

$f$ is one-to-one if and only if $a \neq 0$

$f$ preserves the direction of vectors on the line when $a > 0$ and reverses it when $a < 0$

when $a \neq 0$, the image of an interval of length $l$ is an interval of length $\left|a\right| l$.

the fixed points of the transformation $f$

the transformation inverse to the transformation $f(a \neq 0)$.

Find the image $\mathcal{S}$ of the line under the mapping $f$.

Is the mapping $f$ an injection?

Indicate some sets on the line that are mapped one-to-one onto $\mathcal{S}$.

Find the image $\mathcal{S}$ of the line under the mapping $f$.

Is the mapping $f$ an injection?

Find the preimage $t$ of each point of $\mathcal{S}$.

Is the transformation $f:$

(1) a surjection,

(2) one-to-one?

Find the complete preimage of an arbitrary point $\left(x^{ * }, y^{ * }\right)$ of the plane.

Is the transformation $f$ one-to-one?

Identify regions in the plane on which $f$ is one-to-one.

Let $\bar{f}$ be the restriction of the transformation $f$ to the strip $0 < y < \pi$. Find the formulas defining the mapping inverse to $\bar{f}$.

Find the image of the set $\mathcal{X}$ under the mapping $\varphi : \mathcal{X} \rightarrow \mathcal{X} \times \mathcal{Y}$, defined by the equality $\varphi (x) = (x, f(x))$.

Prove that $f = \pi \varphi$ (for the definition of $\pi$, see 12. 22).

Prove that the mapping $f$ is an injection if and only if $\varphi$ is an injection.

Prove that $f$ is a surjection if and only if $\pi (\Gamma ) = \mathcal{Y}$.

a homothety with center at the point $M_{0}\left(\mathbf{r}_{0}\right)$ and ratio $k \neq 0$

a point reflection (central symmetry) with respect to the point $M_{0}\left(\mathbf{r}_{0}\right)$

a translation by the vector $\mathbf{a}$

an orthogonal projection onto the line $\mathbf{r} = \mathbf{r}_{0} + t\mathbf{a}$

a reflection with respect to the line $\mathbf{r} = \mathbf{r}_{0} + t\mathbf{a}$

a contraction toward the line $\mathbf{r} = \mathbf{r}_{0} + t\mathbf{a}$ with ratio $\lambda > 0$.

if $A$ and $B$ are two distinct fixed points of an affine transformation, then all points of the line $AB$ are fixed as well

if an affine transformation $f$ has a unique fixed point, then all invariant lines (if they exist) pass through this point

the point of intersection of two invariant lines of an affine transformation is fixed.

a homothety with center at the origin and ratio $k$

a homothety with center at the point $M\left(x_{0}, y_{0}\right)$ and ratio $k$

a point reflection with respect to the point $M\left(x_{0}, y_{0}\right)$

a translation by the vector $\mathbf{a}(\alpha , \beta )$.

of the points

(1) $O(0, 0)$;

(2) $E_{1}(1, 0)$;

(3) $E_{2}(0, 1)$;

(4) $E(1, 1)$;

(5) $M(- 1, 5)$;

of the lines

(1) $y = 0$;

(2) $x = 0$;

(3) $x - y + 1 = 0$;

(4) $x - y - 1 = 0$;

(5) $2x + 3y = 7$.

of the points

(1) $O(0, 0)$;

(2) $A(- 1, 2)$;

(3) $B(4, - 5)$

of the lines

(1) $y = 0$;

(2) $x = 0$;

(3) $x + y - 1 = 0$;

(4) $x - y - 1 = 0$;

(5) $x - y + 1 = 0$.

$A(1, 0), B(0, 1), C(1, 1), A^{ * }(- 3, 5), B^{ * }(4, - 3), C^{ * }(0, 0)$

$A(3 / 7, 1), \quad B(1, 1 / 4), \quad C(2, - 1), \quad A^{ * }(- 4, 2), \quad B^{ * }(- 1, 6)$, $C^{ * }(4, 13)$

$A(1, 0), \quad B(- 1 / 2, \sqrt{3} / 2), \quad C(- 1 / 2, - \sqrt{3} / 2), \quad A^{ * } = B$, $B^{ * } = C, C^{ * } = A$

$A(1, 2), B(- 7, 4), C(3, - 6); A^{ * }, B^{ * }, C^{ * }$ are the midpoints of the sides of the triangle $ABC$ opposite the vertices $A, B, C$ respectively.

$A(1, 4), B(- 2, 1), C(0, 3), A^{ * }(0, 0), B^{ * }(1, 0), C^{ * }(0, 4)$

$A(- 2, 0), B(2, - 1), C(0, 4), A^{ * }(- 2, 1), B^{ * }(2, 1), C^{ * }(0, 1)$

$A(2, 0), \quad B(3, - 1), \quad C(4, - 2), \quad A^{ * }(2, 1), \quad B^{ * }(- 2, - 1)$, $C^{ * }(- 6, - 3)$

$A(0, 0), \quad B(- 1, 2), \quad C(1, - 2), \quad A^{ * }(- 1, - 1), \quad B^{ * }(0, 0)$, $C^{ * }(1, 1)$

$x^{ * } = 7x - 3y, \quad y^{ * } = x + y$

$x^{ * } = - 5x + y, \quad y^{ * } = 6x$

$x^{ * } = - 5x + y, \quad y^{ * } = 6x + 1$

$x^{ * } = 2x - y + 3, \quad y^{ * } = - 2x + 2y - 6$

$x^{ * } = 4x + 3y - 1, \quad y^{ * } = - 3x - 2y + 1$

$x^{ * } = x, \quad y^{ * } = y$.

$x^{ * } = 2x + 3y, \quad y^{ * } = - y$

$x^{ * } = - x + y, \quad y^{ * } = x - y$

$x^{ * } = y - 9, \quad y^{ * } = 9x + 1$

$x^{ * } = y, \quad y^{ * } = - x + 1$

$x^{ * } = 2x + y - 3, \quad y^{ * } = 2x + 3y - 6$

$x^{ * } = 5x + 3y + 1, \quad y^{ * } = - 3x - y$

$x^{ * } = 3x - 2y + 5, \quad y^{ * } = 2x - y + 5$.

$x = x^{\prime } + y^{\prime } - 2, \quad y = 2x^{\prime } - y^{\prime } + 3$

$x = 3x^{\prime } - 4y^{\prime } - 5, \quad y = 4x^{\prime } + 3y^{\prime } + 1$.

taking the lines $x - y + 1 = 0, \quad x + y - 1 = 0$ respectively to the lines $3x + 2y - 3 = 0, 2x + 3y + 1 = 0$, and the point $A(1, 1)$ to the point $B(- 1, - 2)$

taking the lines $A_{1} x + B_{1} y + C_{1} = 0$ and $A_{2} x + B_{2} y + C_{2} = 0$ respectively to the $y$-axis and the $x$-axis, and the point $A\left(x_{0}, y_{0}\right)$ to the point $B(1, 1)$ (the point $\mathrm{A}$ does not lie on the given lines).

$x^{2} + y^{2} = 1$

$x^{2} - y^{2} = 1$

$xy = 2$

$y^{2} = - 6x$

$(3x + 4y - 1)(4x - 3y + 1) = 0$

$(x + y - 1)(x + y + 1) = 2$.

$x^{2} + y^{2} = 4$

$\frac{x^{2}}{4} - y^{2} = 1$

$(y + 1)^{2} = 8(x - 1)$

$(x + y - 1)(x - y - 1) = 1$

$(x + 2y - 2)(x + 2y + 2) = 0$.

Write the formulas of an affine transformation of the first kind taking the ellipse $\frac{x^{2}}{4} + y^{2} = 1$ to itself so that the point $A(1, \sqrt{3}/2)$ goes to the point $B(- 2, 0)$.

Solve the same problem for an affine transformation of the second kind.

the point $A_{1}(2, 1)$ goes to the point $B_{1}(4, 4)$, and the point $A_{2}(1$, $1 / 4)$ goes to the point $B_{2}(3, 9 / 4)$

the determinant of the transformation equals 1.

a rotation through angle $\varphi$ about the origin

a rotation through angle $\varphi$ about the point $M\left(x_{0}, y_{0}\right)$

an orthogonal projection onto the $x$-axis

an orthogonal projection onto the line $x - 3y + 1 = 0$

a reflection with respect to the $y$-axis

a reflection with respect to the line $3x + 4y - 1 = 0$

a contraction toward the $x$-axis with ratio $\lambda > 0$

a contraction toward the line $x + y - 2 = 0$ with ratio $1 / 3$

a contraction toward the line $2x - y + 5 = 0$ with ratio 2.

affine

orthogonal?

$x^{ * } = x, \quad y^{ * } = 3y$

$x^{ * } = 2x, \quad y^{ * } = 2y$

$x^{ * } = x - 1, \quad y^{ * } = y + 1$

$x^{ * } = - x, \quad y^{ * } = y$

$x^{ * } = - x, \quad y^{ * } = - y$

$x^{ * } = - y, \quad y^{ * } = x$

$x^{ * } = y, \quad y^{ * } = x$

$x^{ * } = \frac{1}{\sqrt{2}}(x - y), \quad y^{ * } = \frac{1}{\sqrt{2}}(x + y)$

$x^{ * } = \frac{1}{\sqrt{2}}(x + y), \quad y^{ * } = \frac{1}{\sqrt{2}}(x - y)$

$x^{ * } = 3x - 6, y^{ * } = 3y + 2$

$x^{ * } = \frac{1}{2} x - \frac{\sqrt{3}}{2} y - 1, \quad y^{ * } = \frac{\sqrt{3}}{2} x + \frac{1}{2} y + 1$

$x^{ * } = \frac{1}{2} x + \frac{\sqrt{3}}{2} y - 1, \quad y^{ * } = \frac{\sqrt{3}}{2} x - \frac{1}{2} y + \sqrt{3}$

$x^{ * } = - x - 2, \quad y^{ * } = - y + 2$

$x^{ * } = \frac{1}{10}(21x + 12y), \quad y^{ * } = \frac{1}{10}(12x + 14y)$

$x^{ * } = \frac{1}{3}(2x + y - 2), \quad y^{ * } = \frac{1}{3}(x + 2y + 2)$

from problem 12.40(3): the affine transformation of the plane taking the points $A, B, C$ respectively to $A^{ * }, B^{ * }, C^{ * }$, $A(1, 0), \quad B(- 1 / 2, \sqrt{3} / 2), \quad C(- 1 / 2, - \sqrt{3} / 2), \quad A^{ * } = B$, $B^{ * } = C, C^{ * } = A$

from problem 12.41(2): the linear transformation of the plane taking the points $A, B, C$ respectively to $A^{ * }, B^{ * }, C^{ * }$, $A(- 2, 0), B(2, - 1), C(0, 4), A^{ * }(- 2, 1), B^{ * }(2, 1), C^{ * }(0, 1)$

the images of the points $O(0, 0)$ and $B(1, 0)$

the preimages of the points $O$ and $B$

the images of the lines $x = 0$ and $y = x$

the preimages of the lines $y = 0$ and $y = - x$.

is parallel to the $x$-axis

is tangent to the circle $x^{2} + y^{2} = 25 / 2$?

the area of a parallelogram whose sides are given by the equations $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0, a_{1} x + b_{1} y + d_{1} = 0, a_{2} x + b_{2} y + d_{2} = 0$

(р) the area of a triangle whose sides are given by the equations $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0, a_{3} x + b_{3} y + c_{3} = 0$.

symmetric to the given circle with respect to the line $x + y - 6 = 0$

obtained from the given circle by a rotation through angle $\operatorname {arctg}(4 /3)$ about the origin;

obtained from the given circle as a result of a homothety with center $A(6, 0)$ and ratio 4.

$f: x^{ * } = y, y^{ * } = x; g: \dot{x}^{ * } = 3x + 4y + 1, y^{ * } = - 7x + 5y - 2$

$f: x^{ * } = 4x - 2y + 6, y^{ * } = - 3x + y; g: x^{ * } = x - y, y^{ * } = 4x + y + 1$.

$f$ is a translation by the vector $\mathbf{a}(- 1, 1); g$ is a homothety with center at the point $M(1, 2)$ and ratio 3

$f$ is a homothety with center at the point $M(2, - 1)$ and ratio $- 1 / 2; g$ is a point reflection with respect to the point $N(3, 1)$.

$x^{ * } = y + 3, y^{ * } = - x + 3y - 1$

$x^{ * } = 3x + 4y + 8, y^{ * } = 5x + 7y + 6$

$x^{ * } = \frac{4}{5} x + \frac{3}{5} y + \frac{1}{5}, y^{ * } = \frac{3}{5} x - \frac{4}{5} y - \frac{2}{5}$

$x^{ * } = 3x + 5y - 4, y^{ * } = 5x + 9y + 6$

$x^{ * } = 3x - 24, y^{ * } = - x + 4y + 12$

$x^{ * } = 2x - y, y^{ * } = - 4x + 2y$

$x^{ * } = 4x - 3y, y^{ * } = 3x + 4y$