Math Problems Vault

Chapter IV

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Section IV.1§ 1[3/100%]

Problem IV.1.1

Draw three crosses and four noughts.

(a)

How should the arrows go from the crosses to the noughts so that a map of the set of crosses into the set of noughts is obtained? Suppose the arrows define a map.

(b)

Can they be drawn so that every image has a unique preimage?

(c)

Can they be drawn so that every nought has a preimage?

(d)

Answer the same questions if there are four crosses and three noughts.

(e)

For what number of noughts is a one-to-one map of a set of three crosses possible?

Problem IV.1.2

Let the transformations $\mathrm{f}$ , $\mathrm{g}$ and $\mathrm{h}$ have inverses. Find the transformation inverse to their product $\mathrm{fgh}$ .

Problem IV.1.3

Write the formulas defining the axial symmetry about the line having the equation $x+y=5$ in a Cartesian rectangular coordinate system.

Section IV.2§ 2[13/100%]

Problem IV.2.1

Are the transformations defined by the formulas affine: а)

\begin{aligned} & x^{*}=x+y-1, \\ & y^{*}=x-y+1, \end{aligned}

б)

\begin{aligned} & x^{*}=x-y-1 \\ & y^{*}=-x+y+1 \end{aligned}

Problem IV.2.2

Find the image of the line $x-y=2$ under the transformation (а) of Problem 1.

Problem IV.2.3

Prove, without resorting to coordinate notation, that an orthogonal transformation is one-to-one.

Problem IV.2.4

A point $A$ is called a fixed point of a transformation $\mathrm{f}$ , if $\mathrm{f}(A)=A$ . Find the fixed points of the transformation (а) of Problem 1.

Problem IV.2.5

Prove that a linear transformation which is not the identity either has one unique fixed point, or has a line consisting of fixed points, or has none at all.

Problem IV.2.6

How will the formulas defining a linear transformation change if the origin of coordinates is moved to a fixed point without changing the basis vectors?

Problem IV.2.7

(a)(b)

A linear transformation in the coordinate system $O, \mathbf{e}_{1}$ , $\mathrm{e}_{2}$ is given by the formulas

x^{*}=a_{1} x+b_{1} y+c_{1}, \quad y^{*}=a_{2} x+b_{2} y+c_{2} .

By what formulas is it given in the coordinate system

(a)

$O$ , $\mathbf{e}_{2}, \mathbf{e}_{1}$

(b)

$O, \mathbf{e}_{1}, 2 \mathbf{e}_{2}$ .

Problem IV.2.8

Prove that the linear transformation given in a Cartesian rectangular coordinate system by the formulas

x^{*}=x \cos \varphi +y \sin \varphi , \quad y^{*}=x \sin \varphi -y \cos \varphi ,

is an axial symmetry. Find the equation of the axis of symmetry.

Problem IV.2.9

Can it happen that the product of two linear transformations is affine if one of them is not affine?

Problem IV.2.10

Let an affine transformation in a Cartesian rectangular coordinate system be given by the formulas

x^{*}=x+b y+c_{1}, \quad y^{*}=a x+c_{2} .

Find the vectors orthogonal to their images.

Problem IV.2.11

A triangle with vertices $A(1,0), B\left(-\frac{1}{2}, 1\right)$ and $C\left(-\frac{1}{2},-1\right)$ is given. Find the linear transformation carrying each vertex into the midpoint of the opposite side.

Problem IV.2.12

Prove that the transformation of Problem 8 is the product $\mathrm{gf}$ of the axial symmetry $\mathrm{f}$ about the axis of abscissas and the rotation $\mathrm{g}$ through the angle $\varphi$ about the origin of coordinates. What transformation is obtained if $\mathrm{f}$ and $\mathrm{g}$ are multiplied in the other order?

Problem IV.2.13

Let an affine transformation have one unique fixed point. Prove that in this case every invariant line (if such lines exist) passes through this point.

Section IV.3§ 3[8/100%]

Problem IV.3.1

Find the area of a triangle if its sides lie on the lines with the equations $x+y=1, x-y=-1$ and $2 x+y=2$ in a Cartesian rectangular coordinate system.

Problem IV.3.2

Suppose that under an affine transformation the points $A, B$ and $C$ are carried into the points $A^{*}, B^{*}$ and $C^{*}$ . Prove that the point of intersection of the medians of $\triangle A B C$ is carried into the point of intersection of the medians of $\triangle A^{*} B^{*} C^{*}$ .

Problem IV.3.3

(a)(b)

We shall say that an affine transformation stretches a vector $\mathbf{a}$ by a factor of $\alpha$ if $\left|\mathbf{a}^{*}\right|=\alpha \left|\mathbf{a}\right|$ . For the transformation given in a Cartesian rectangular coordinate system by the formulas

x^{*}=4 x+7 y, \quad y^{*}=8 x+y,

find the vectors for which the stretching is

(a)

maximal

(b)

minimal.

Problem IV.3.4

Suppose a line is tangent to a second-order curve. Prove that under an arbitrary affine transformation the image of the line is tangent to the image of the curve.

Problem IV.3.5

Prove that the vertices of a rhombus circumscribed about an ellipse lie on its axes of symmetry.

Problem IV.3.6

(a)(b)

Represent as a product of two axial symmetries

(a)

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

(b)

the rotation through the angle $\varphi$ about the point $O$ .

Problem IV.3.7

Represent the compression toward the axis of abscissas of a Cartesian rectangular coordinate system as a product of the translation by $\mathbf{a}(0, a)$ and a compression toward another line.

Problem IV.3.8

Can the product of two rotations of the plane be a translation?