Vocabulary
1. A list of words to remember:
coincident – совпадающий, совмещенный
continuous - непрерывный
to converge – сходиться, стремиться (к пределу)
corresponding – соответствующий
fixed point – базисная/ неподвижная/ фиксированная точка
Given… - если дано…
invertible – обратимый элемент, поддающийся преобразованию
inverse – обратное (величина, функция, процесс…)
to map smth to smth – отображать что-то на что-то
one-to-one – взаимно однозначное
to rescale – изменять масштаб
Make your own sentences on the topic using the words above.
2. Fill in the gaps in the sentences with these words.
1) The equation of the … image can be written in the form (x2+y2)2=2(x2-y2)+k-1.
2) … three distinct points in the extended plane, there exists a linear transformation T which carries these points into 0,1,∞.
3) The … of the linear transformation is linear.
4) The existing of an … shows that the correspondence defined by T is …
5) The best way to do this is to use the appropriate linear transformation to … one variable so that its mean and its SD match the other variable's.
6) In mathematics, a … (also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.
7) In this paper we show that certain linearly interacting diffusions … to super Brownian motion if suitably rescaled in time and space.
Read Text 2 and report on this type of linear transformations.
Text 2.
8.1. Euclidean Linear Transformations
By a
transformation from Rn
into Rm,
we mean a function of the type T: Rn→Rm,
with domain Rn
and codomain Rm.
For every vector x
Rn,
the vector T(x)
Rm
is called the image of x under the transformation T , and the set
R(T) ={T(x) : x Rn},
of all images under T, is called the range of the transformation T.
Remark.
For our convenience later, we have chosen to use R(T) instead of the usual T(Rn) to denote the range of the transformation T.
For every x= (x1 …,xn) Rn, we can write T(x) =T(x1 ,…,xn) = (y1 ,…,ym). Here, for every i= 1, …,m, we have
yi=Ti(x1,…,xn), (1)
where Ti:Rn→R is a real valued function.
Definition.
A transformation T:Rn→Rm is called a linear transformation if there exists a real
matrix A=
…
such that for every x= (x1,…,xn) Rn, we have T(x1,…,xn) = (y1,…,ym), where
y1=a11x1+… +a1nxn;
.
.
.
ym=am1x1+…+amnxn; (2)
which can also be written in matrix notation.
The matrix A is called the standard matrix for the linear transformation T.
Remarks.
(1) In other words, a transformation T:Rn→Rm is linear if the equation (1) for every
i = 1,…,m is linear.
(2) If we write x Rn and y Rm as column matrices, then (2) can be written in the form y=Ax, and so the linear transformation T can be interpreted as multiplication of x Rn by the standard matrix A.
Definition.
A linear transformation T:Rn→Rm is said to be a linear operator if n=m. In this case, we say that T is a linear operator on Rn.
Example 8.1.1.
The linear transformation T:R5→R3, defined by the equations
y1= 2x1+3x2+5x3+7x4-9x5,
y2= 3x2+4x3 +2x5,
y3= x1 +3x3-2x4,
can be expressed in matrix form as
.
If (x1 ,x2, x3, x4, x5) = (1,0,1,0,1), then
,
so that T(1,0,1,0,1)=(-2,6,4).
Example 8.1.2.
Suppose that A is the zero m x n matrix. The linear transformation T:Rn→Rm,
where T(x) =Ax for every x Rn, is the zero transformation from Rn into Rm. Clearly T (x) =0 for every x Rn.
Example 8.1.3.
Suppose that I is the identity n x n matrix. The linear operator T:Rn→Rn, where T(x) =Ix for every x Rn, is the identity operator on Rn. Clearly T(x) = x for every
x Rn.
PROPOSITION 8A.
Suppose that T:Rn→Rm is a linear transformation, and that {e1,…,en} is the standard basis for Rn. . Then the standard matrix for T is given by A=(T(e1),…,T(en) ), where T(ej) is a column matrix for every j = 1,…,n.
Proof.
This follows immediately from (2).