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Lecture 10

LINEAR TRANSFORMATIONS

Just to confuse you, a linear transformation is also
called a linear function, or a linear map, or a
linear operator, or a linear transform. 
However, the terms operator and transform are
usually used when the vectors are functions. 

2

Definition
The transformation f : U  V , where U and V are linear spaces

over the same field K, is called a linear transformation if the
following equivalence holds for arbitrary vectors v

1

, v

2

 U and

arbitrary constant c  R

f( v

1

+ v

2

) = f(v

1

) + f(v

2

) - addivity

f(c v) = c f(v) - homogeneity

Examples :

1. f: R → R, f(x) = 6x
2. f: R

2

→ R

2

, f(x, y) = (x+y, 2x)

3. f: R

3

→ R

2

, f(x, y, z) = (x+4y, x+z)

3. f: R

2

→ R

3

, f(x, y) = (x+y, x, 2y)

The two above conditions are equivalent
to one

f(c

1

v

1

+ c

2

v

2

) = c

1

f(v

1

) + c

2

f(v

2

).



3

Let f be a linear transformation of V, then

f (0) = f (0 v) = 0 f(v) thus f (0) = 0

Hence, the image of the vector 0 is 0.

Image of the
vector 0.

f(0) = 0

Proof

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Examples of Some Linear Transformations

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Rotation around a centre
(point) of rotation (0,0)

Projection on line P(v)

)

cos

y

sin

x

,

sin

y

cos

x

(

)

y

,

x

(

Q















)

y

,

x

(

u 

1.

2.

6

Rotation around the centre

7

8

Homothety

Homothety, dilation, central similarity are all interchangeable
terms used to describe a geometric transformation defined by a
point O called the center of homothety and a real number k,
known as its coefficient or ratio..

3.

Homothety with centre (0,0) and ratio k, then f: R

2

→ R

2

, f(x,y) = (kx,ky)

9

Reflection about the xy-plane

x

y

z

4.

R(x,y,z) = (x,y,-z)

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3. There is a fixed scale, a level of magnification, for
any given line.  If a centimeter of line is expanded to a
kilometer under the transformation, then every cm of
that line expands to a km.  Of course other lines
(independent vectors) may expand by a different
amount, or they may be squashed down to a point
(expanded by 0).

2. Since the image of k times a vector is k times the
image of the vector, a linear transformation maps
lines to lines in real space or to zero i.e. squashes
them into the origin. 
 

1. The linear transformation does not change the
proportionality of vectors. If y = cx then f (y) = c f(x),
which means that the images of proportional vectors
are proportional with the same proportionality
constant.

Similarity Transformations

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Matrix of Linear Transformation

12

Let the basis B

1

of linear space U be composed of vectors: e

1

, e

2

, ..., e

n

and the basis B

2

of V be composed of: h

1

, h

2

, ..., h

m

.

The basis vectors of U are transformed into: f (e

1

) = w

1

, f (e

2

) = w

2

, ..., f (e

n

) = w

n

.

n

n

e

v

e

v

e

v

v















2

2

1

1

The coordinates of w

j

relative to B

2

are denoted by [ w

1j

, w

2j

, ..., w

mj

]

MATRIX OF LINEAR TRANSFORMATION f: U  V

.

h

w

h

w

h

w

w

m

mj

j

j

j











2

2

1

1

 

 

 





























n

n

n

n

w

v

w

v

w

v

e

f

v

e

f

v

e

f

v

)

v

(

f





2

2

1

1

2

2

1

1

 































m

i

i

n

j

j

ij

m

i

i

ij

n

j

j

.

h

v

w

h

w

v

1

1

1

1

then v is carried into

An arbitrary vector v  U is expressed as a linear combination of the

basis vectors B

1

:

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So the coordinates of f ( v ) relative to the basis B

2

of space

V

are

,

v

v

w

w

w

w

n

mn

m

n





















































1

1

1

11

where the coordinates of vectors w

1

, w

2

, ..., w

n

relative to

B

2

, are written in columns.























mn

m

n

f

w

w

w

w

A











1

1

11

is called the transformation matrix of f.

DEFINITION

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Theorem

The rank of the transformation matrix A

f

is equal to

the dimension of Im(f ):
Rank A

f

= dim Im( f ).























mn

m

n

f

w

w

w

w

A











1

1

11

If the bases of linear spaces are not specified it means that the unit bases are used.

15

























n

f

v

v

A

v

f



1

)

(

where [ v

1

, v

2

, ..., v

n

] are the coordinates of v relative to the B

1

basis.

How to build the transformation matrix.
Let f denote the linear transformation.

Step 1. Find the images of the basis vectors from B

1

f (e

1

) = w

1

, f (e

2

) = w

2

, ..., f (e

n

) = w

n

.

Step 2. Find the coordinates of w

1

, w

2,

...,w

n

relative to the

basis vectors from B

2

Step 3. The matrix A

f

is formed by writing the coordinates of

w

i

in the i-th column of the matrix.

Then the coordinates of vector f ( v ) relative to the B

2

basis are

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The linear transformation is uniquely defined by the
matrix relative to given bases of U and V.

The properties of a linear transformation can be interpreted in
terms of the matrix of the transformation.

If A is a real (m x n) matrix, then the rule f(x) = Ax describes a
linear transformation R

n

→ R

m

One linear transformation may be represented by many
matrices.
This is because the values of the elements of the matrix depend
on the bases that are chosen.

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Examples:
1. Unit bases
2. Arbitrary bases

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Some special cases of linear transformations of
two-dimensional space R

2

:

•rotation by 90 degrees counterclockwise:

•

 

             

•reflection against the x axis:

 

             

•scaling by 2 in all directions:

 

           

•projection onto the y axis:     

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Kernel and Range

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Definition

Kernel of a linear transformation
The kernel (null-space) of a linear transformation f: V
→ W is the set of all vectors v such that f(v) = 0:
Ker(f) = {v V : f(v) = 0}

Fact

The kernel is a subspace of V

Since f(0) = 0, the kernel is not empty.
If u an v are in the kernel, then
f(r u + s v) = r f(u) + s f(v) = r 0 + s 0
= 0
So, r u + s v is in the kernel. Hence the kernel is a
subspace of V.



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Definition
The image (or range) of a transformation f : V → W:
Im (f) = {f(v): v V}

Fact

The image is a linear subspace of W.



22

Projection on a space

23

The projection of a vector on the subspace spanned by vectors

v

1

, ..., v

k

 R

n

.

Let v

1

, ..., v

n

be the basis of R

n

and c

1

, c

2

, ..., c

n

be the coordinates of

u relative to this basis. The projection on lin { v

1

, ..., v

k

} has the

following form:

f(u) = c

1

v

1

+ c

2

v

2

+ ... + c

k

v

k

.

The transformation matrix relative to the basis { e

1

, e

2

, e

3

} has the following form:

.

A

f























0

0

0

0

1

0

0

0

1

Definition

Example
The projection of R

3

on Lin {e

1

,e

2

}.

Then v

1

= e

1

= (1, 0, 0), v

2

= e

2

= (0, 1, 0), any vector v = (a, b, c)

is mapped as
f( v) = a e

1

+ b e

2

= (a, b, 0).

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Sum and Product

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Linear transformations can be added to one another
and composed.

Definition
If f, g : U  V are linear transformations, then their sum is

defined as follows:
( f + g )(u) = f(u) + g(u). for uU

Definition
If f : U  V and g : V W are linear transformations, then their

composition is defined as follows:

f  g ( u ) = g( f ( u )) for uU.

Fact

The sum and composition of linear transformations is
a linear transformation.

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MORE EXAMPLES

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(3,2)

(0,0)

(0,3)

Pixel Coordinates

y

x

x =
-0.5

y = 4.5

y = -03.5

x = 4.5

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Examples in 2D
graphics

Rotation
For rotation by an angle θ counterclockwise about
the origin, the functional form is x' = x cosθ − y
sinθ and y' = x sinθ + y cosθ. Written in matrix
form, this becomes:

 

                           

Scaling
For scaling (that is, enlarging or shrinking), we have
x’ = s

x

x and y’ = s

y

y. The matrix form is

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Shearing

For shearing there are two possibilities. A shear
parallel to the x axis has x' = x + ky and y' = y; the
matrix form:

A shear parallel to the y axis has x' = x and y' = y + kx,
which has matrix form:

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Orthogonal projection

To project a vector orthogonally onto a line that goes
through the origin, let

(

u

x

, u

y

) be a unit vector in the

direction of the line. Then use the transformation
matrix

A reflection about a line that does not go through the
origin is not a linear transformation; it is an affine
transformation

.

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y

x

y

x

(a, b)

This is not a linear transformation. The origin
moves.

f(x, y) = (x + a, y
+ b)

Caution !!!

Translation

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As with reflections, the orthogonal projection onto a
line that does not pass through the origin translation
is

not linear

.

Perspective projections are also

not linear

.

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x

x

y

z

y

Embed the xy-plane in R

3

at z

= 1.

(x, y)  (x, y, 1)

How do we manage to represent the above transformations
by a matrix??

NEW COORDINATES

(homogeneous coordinates)

34

2D Linear Transformations as 3D
Matrices

11

12

11

12

21

22

21

22

a

a

a x a y

x

a

a

a x a y

y

+

�

�

�

�

��

=

�

�

�

�

��

+

��

�

�

�

�

Any 2D linear transformation can be represented by a 2x2
matrix.

11

12

11

12

21

22

21

22

0
0

0

0 1 1

1

a

a

x

a x a y

a

a

y

a x a y

+

�

��� �

�

�

��� �

�

=

+

�

��� �

�

�

��� �

�

�

��� �

�

or a 3x3 matrix

we use a „dummy dimension”

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2D Linear Translations

as 3D Matrices

Any 2D translation can be represented by a 3x3 matrix.

1 0
0 1
0 0 1 1

1

a x

x a

b y

y b

+

�

��� �

�

�

��� �

�

=

+

�

��� �

�

�

��� �

�

�

��� �

�

This is a 3D shear that acts as a translation on the plane
z = 1.

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y

x

(a, b)

(x, y)(x+a,

y+b)

Translation

37

Translation as a Shear

x

x

y

z

y

38

Summary
To represent not linear (affine transformations ) with matrices,
we must use homogeneous coordinates. This means representing
a 2-vector (x, y) as a 3-vector (x, y, 1), and similarly for higher
dimensions. Using this system, translation can be expressed with
matrix multiplication. The functional form x' = x + t

x

; y' = y + t

y

becomes:

                         

All ordinary linear transformations can be converted into affine
transformations by expanding their matrices by one row and
column, filling the extra space with zeros except for the lower-
right corner, which must be set to 1. For example, the rotation
matrix from above becomes:

                    

Using this system, translations can be intermixed with all other
types of transformations, using the same matrix multiplication as
before.

Translation in real space can be represented as a shear in real
projective space.)

a "dummy dimension"

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MERRY CHRISTMAS AND A HAPPY NEW YEAR !