Math Problems Vault

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}

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Problem IV.2.2

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

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Problem IV.2.3

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

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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.

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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.

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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?

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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

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(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.

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Problem IV.2.9

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

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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.

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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.

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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?

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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.

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