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# affine transformation

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Theorem 147) suppose that each of V and W is a linear space; A is an affine transformation
from V to W, a is a number,; and each of x and y is a point of V. Then
( )
( )
( )
1 1A ax a y aAx a Ay+ = +
Proof: recollect the definition; suppose that n is a positive integer, each of
1 2
, ,...,
n
y y y
is a
point of the linear space V, and x is a point of V. The statement, “x is an affine combination of
{ }
1 2
, ,...,
n
y y y
means that there is a number-sequence
1 2
, ,...,
n
a a a
such that
1
1
n
i
i
a
=
=
and
1
n
i i
i
x a y
=
=
In view of this definition, suppose x and y are in the finite dimensional linear space V.
Then
1 1 2 2
... ,
n n
x a y a y a y= + + +
and
1 2
... 1
n
a a a+ + + =
Similarly,
1 1 2 2
... ,
n n
y b y b y b y= + + +
and
1 2
... 1
n
b b b+ + + =
So, we consider
( ) ( ) ( )
1 1 2 2 1 1 2 2
... 1 1 ... 1
n n n n
aa y aa y aa y a b y a b y a b y= + + + + + + +
( )
{ }
( )
{ }
( )
{ }
1 1 1 1 2 2 2 2
1 1 ... 1
n n n n
aa y a b y aa y a b y aa y a b y= + + + + + +
( )
{ }
( )
{ }
( )
{ }
1 1 1 2 2 2
1 1 ... 1
n n n
aa a b y aa a b y aa a b y= + + + + + +
( )
{ }
{ } ( )
{ }
( )
1
1 1
1 1
1
1
1
n
i i i
i
n n
i i i i
i i
n n
i i i i
i i
aa a b y
aa y a b y
a a y a b y
=
= =
= =
= +
= +
= +
Actually, we can write this line directly also and not needed to beat round the bush.
But, the addition of vectors and scalar multiplication i wished to exhibit.
Apply A on both sides
( )
( )
( )
1 1
1 1
n n
i i i i
i i
A ax a y A a a y a b y
= =
+ = +
÷
Recollect that an affine transformation is also a linear transformation which preserves the
constants also.
So, we can write the right side of the above equation as
( )
1 1
1
n n
i i i i
i i
A a a y A a b y
= =
= +
÷ ÷
By the addition of image vectors
( )
1 1
1
n n
i i i i
i i
aA a y a A b y
= =
= +
÷ ÷
By the scalar multiplication of image vectors
( )
1aAx a Ay= +
By the definition of x and y.

Observe that
( )
1 1a a+ =
which satisfies the important part of the definition of affine
combination of x and y.
If this condition does not hold, then the above equality is not possible.
148) if each of V and W is a linear space, A is an affine transformation from V to W, and x
and y are two points of V, then
{ }
( )
{ }
, ,A Line x y Line Ax Ay=
Proof: recollect the definition of the line in a linear space.
Suppose x and y are two points in a linear space V. The affine span of
{ }
,x y
, denoted by
{ }
,AS x y
is the set
( )
{ }
1 : is a numberax a y a+
If
x y
, then
{ }
( )
,AS x y
is called the line through x and y (or containing x and y) is denoted
by
{ }
,Line x y
We are required to prove that the affine transformation A preserves the line joining x and y.
Observe that if
0 1a
, then any point z such that
x z y
can be written as
( )
1z ax a y= +
by the elementary algebra.
In fact, z is the locus of a line joining x and y that exists between x and y.
If a > 1 or a < 0 holds, then the point z lies away from x and y which can be seen as a point on
the line that contains x and y.
Now, apply A on both sides,
( ) ( )
( )
1A z A ax a y= +
By the theorem 147, it can be written as
( ) ( )
1A z aAx a Ay
= +
Since z is an arbitrary point on the line
{ }
,Line x y
, the above application is true for every
point on this line. So, we can write
{ }
( )
{ }
, ,A Line x y Line Ax Ay=
Theorem 150) suppose that each of V and W is a linear space, and that A is a transformation
from V to W with the property that, if a is a number and x and y are two points in V such that
{ }
( )
{ }
, ,A Line x y Line Ax Ay=
, then A is an affine transformation.
That means, this is the converse of the mixture of 147 and 148.
Proof: given that
{ }
( )
{ }
, ,A Line x y Line Ax Ay=
Also, if x is in the linear space V, then it can be written as
1 1
...
n n
x a y a y= + +
for some points
1 2
, ,...,
n
y y y
of V and
1 2
, ,...,
n
a a a
are numbers such that
1 2
... 1
n
a a a+ + + =
Similarly,
1 1
...
n n
y b y b y= + +
and
1 2
... 1
n
b b b+ + + =
Further, it is easy to see that the linear combination of the basis
1 2
, ,...,
n
y y y
of V is again a
point in V.

So, x and y are distinct points of V provided
i i
a b
for at least one
,1i i n
Considering the distance between x and y as the norm and is equated to 1, then any point
between x and y will be a distance ratio of
:1a a
from x to y and thus, that point also lies in
V.
A preserves all such points given by
{ }
( )
{ }
, ,A Line x y Line Ax Ay=
So, we can write that
( )
( )
( )
1 1A ax a y aAx a Ay+ = +
In other words, A preserves lines and shifts.
Thus, A is an affine transformation.

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