3. Translate the following text into English.

Преобразование устанавливает соответствие между числами или точками. Оно ни коим образом не отличается от функции, но мы применяем термин «преобразование», когда хотим передать идею активной замены одной точки другой. Простым примером преобразования является параллельное смещение комплексной плоскости, которое заменяет точку z точкой z+a. Мы можем рассматривать точки z и z+a как лежащие в одной и той же или разных плоскостях. В первом случае подходящим обозначением было бы z → z+a; в последнем случае мы бы применили функциональное обозначение w=z+a. В более общем виде, если преобразование обозначено через Т, то мы будем писать либо z→Tz, либо w=Tz.

Двойное (ангармоническое) отношение.

Если на расширенной плоскости даны три разные точки z₂,z₃,z₄, то существует линейное преобразование Т, которое переводит эти точки в 0,1,∞. Если ни одна из данных точек не является ∞, то Т будет задано формулой

Tz= .

Если z₂,z₃ или z₄=∞, то преобразование, соответственно, приводится к виду

.

Если бы S было другим преобразованием с тем же самым свойством, тогда ST^-1 оставило бы точки 0,1,∞ инвариантными. Для

ST^-1=

эти условия означают, что b=c=0, a=d.

Следовательно, ST^-1 было бы приведено к тождественному преобразованию, и мы бы имели S=T. Отсюда заключаем, что Т определяется единственным образом.

Problem solving

Can you prove the following propositions and explain how you obtained the results?

PROPOSITION 1.

Suppose that T₁:Rⁿ→R^m and T₂:R^m→R^k are linear transformations. Then T=T₂ ○ T₁:Rⁿ→R^k is also a linear transformation.

PROPOSITION 2.

Suppose that the linear operator T:Rⁿ→Rⁿ has standard matrix A. Then the following statements are equivalent:

(a) The matrix A is invertible.

(b) The linear operator T is one-to-one.

(c) The range of T is Rⁿ, in other words, R(T) =Rⁿ.

PROPOSITION 3.

A transformation T:Rⁿ→R^m is linear if and only if the following two conditions are satisfied:

(a) For every u,v Rⁿ, we have T(u+v) =T(u) +T(v).

(b) For every u Rⁿ, and c R, we have T(cu) =cT(u).

*Source: WWL Chen, Linear Algebra. Chapter 8 Linear Transformations 2008.

http://rutherglen.science.mq.edu.au/wchen/lnlafolder/la08.pdf

Grammar Notes:

6. The Gerund and its forms

The forms of gerund coincide with the forms of participle 1: Ving (*the difference is in functions):

	Active	Passive
Simple	solving	being solved
Perfect	having solved	having been solved

In fact, the use of the perfect gerund, both active and passive, is very often redundant, as the time reference is usually clear from the context:

They reported finding the solution of this problem.

With a number of verbs we often tend to find a perfect gerund, although a simple gerund would be sufficient. These verbs include admit, celebrate, deny, mention, recall. Thus, They denied having seen him and They denied seeing him are equally possible.

The translation into Russian usually depends on the meaning and context; very often the gerund is translated with a subordinate clause (especially the passive and perfect forms), but it also can be translated with an infinitive, a participle, a noun with a preposition, etc.:

There is no necessity of making conclusions right now.

Нет никакой необходимости делать выводы прямо сейчас.

He easily reduced equation (17) to the form (18) without using Green's theorem.

Он легко привел уравнение (17) к виду (18), не применяя (без применения) теорему Грина.

Besides being extremely simple, this method is very popular.

Не говоря уже о том, что этот метод очень прост, он очень популярен.

We look forward to much attention being given to this investigation.

Мы ожидаем, что этому исследованию будет уделено большое внимание.

The gerund can be used singly (He was intent on winning), in a phrase after a preposition (She left without saying a word), or in a construction with a noun or pronoun performing this action (It resulted from the experiment being incomplete).

In a gerund construction, the gerund is in predicative relation to a noun or a pronoun. The subject of the sentence is not identical with the subject performing the action denoted by the gerund. The noun can be used in the common case or in the possessive form (in formal style):

This terminology needs the professor/the professor’s improving.

A pronoun should be either a possessive one or a personal pronoun in the objective case: I dislike his/him coming here every day.

I insisted on my/me being examined by our lecturer in algebra.

She insisted on both of them being wrong.

The gerund differs from the participle 1 in being used after prepositions; in performing the functions of the Subject and an Object and emphasizes the action or the state of an object instead of characterizing it (as the participle 1 does).

Remember the prepositions after which we usually use the gerund:

on the account of – из-за;

apart from/aside from – помимо, кроме;

in addition to – кроме, вдобавок;

besides – кроме;

because of – из-за, по причине;

despite/in spite of – несмотря на;

due to – благодаря, вследствие;

except(for) – кроме, за исключением;

instead of – вместо;

owing to – благодаря;

with the object/view to/ with the aim of – с целью;etc.

Exercise 1. Translate the following sentences into Russian using a subordinate clause and paying attention to the prepositions used.

1. Instead of being moved to the right, the dot is moved to the left.

2. We look forward to much attention being given to the use of this approach.

3. In our example we referred to the probability of female’s being colourblind instead of saying “the conditional probability of a randomly chosen person’s being colourblind” on the hypothesis that the person is a female.

4. All the evidence of our experiments is against such statement being true.

5. In addition to describing in detail how to operate with these numbers, he proved a number of theorems.

6. The game took about 2 hours due to explaining all the rules and just general first time learning the new system.

7. Besides observing the changing planet, the Earth observing satellite Landsat 5 recorded and took images of many events in different cities.

8. Despite explaining things numerous times, we were not getting far.

9. We exhibit an example and a proof concerning these facts after giving a slightly simpler system.

10. The simple things become great because of proving underlying phenomena by research.

Exercise 2. Complete the sentences using the gerund or gerund constructions.

1. There are several ways of (решить эту проблему).

2. We shall give only one example here for (вычислить эту величину).

3. (стоит отметить) that none of these questions depends on the uniform algebra setting.

4. Finally the studies covered are only skeleton of actual studies sufficient for the purpose of (демонстрировать эту методику).

5. The chapter also discusses, at some length, the problem of (составлять программу и оценить результаты).

6. Theorem 10.2 can be proved without (использовать Лемму 5.1) from this paper.

7. After (завершить это исследование) the authors learned of the report by Hill and Lee, which deals with the initial stages of the analogous problem.

8. By (применять метод математической индукции) using Laplace transformation and its inverse it is not difficult to derive a formal extension of formula (2.16) in the form (3.1).

9. The main part of the paper consists in (определить все неразложимые объекты в К).

10. In the paper under review by (применять метод) employed in the previous papers, the author proves several important results.