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affine transformation theorems

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a coincidence that i could click to answer. Otherwise, somebody might have taken it and i
missed the chance.
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.

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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.

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Thank you for posting this question and god has given me this opportunity. Because, it is just a coincidence that i could click to answer. Otherwise, somebody might have taken it and i missed the chance. 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 Proof: recollect the definition; suppose that n is a positive integer, each of is a point of the linear space V, and x is a point of V. The statement, "x is an affine combination of" means that there is a number-sequence such that and In view of this definition, suppose x and y are in the finite dimensional linear space V. Then and Similarly, and So, we consider 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 e ...
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