Linear Transformation: Definition and Examples

1. 1.9 Linear Transformations
Dr. Farhana Shaheen

2. Objectives:
• Learn to view a matrix geometrically as a function.
• Learn examples of matrix transformations: reflection, dilation, rotation,
shear, projection.

3. MATRIX OF LINEAR
TRANSFORMTION
• Transformation is a process of modifying and re-positioning the existing
graphics.
• Transformations are helpful in changing the position, size, orientation, shape
etc of the object.

4. Geometry of Linear Transformations
• https://math.hmc.edu/calculus/hmc-mathematics-calculus-online-
tutorials/linear-algebra/geometry-of-linear-transformations/

5. 2D Transformations
• 2D Transformations take place in a two dimensional plane.
• 2D Shearing is an ideal technique to change the shape of an existing object
in a two dimensional plane.

6. In computer graphics, various transformation
techniques are-

7. Shear Transformation
•
•  Stretched along x-axis (Horizontal Shear)
•
•  Stretched along y-axis (Vertical Shear)






10
01






10
1 s






10
01






1
01
t

10. Rotation at an angle θ

11. Rotation at an angle θ
• The standard matrix for the linear transformation that rotates vector by an
angle θ is
•








cossin
sin-cos

12. Rotation Matrix

13. Problem-01:
Given a line segment with starting point as (0, 0) and ending point as (4, 4).
Apply 30 degree rotation anticlockwise direction on the line segment and find
out the new coordinates of the line.

15. REFLECTION
For every line in the plane, there is a linear transformation
that reflects vectors about that line.
Reflection about the x-axis and y-axis is given by the
standard matrices:

































y
x
y
x
Tto
10
01
10
01
































y
x
y
x
Tto
10
01
10
01

16. Reflection through x-axis
• Point in 1st Quadrant will be reflected in 4th Quadrant:

































y
x
y
x
Tto
10
01
10
01

17. Reflection through y-axis
• Point in 1st Quadrant will be reflected in 2nd Quadrant:
































y
x
y
x
Tto
10
01
10
01

18. Reflection through the line y = x
• Point in 2nd Quadrant is reflected in 4th Quadrant:
































x
y
y
x
Tto
10
01
10
01

19. Reflection through the line y = -x
• Point in 1st Quadrant is reflected in 3rd Quadrant:




































x
y
y
x
Tto
01
10
10
01

20. Rotation through an angle of -135 degrees
then reflected through x-axis

21. Problem-02:
Given a triangle with corner coordinates (0, 0), (1, 0)
and (1, 1). Rotate the triangle by 90 degree
anticlockwise direction and find out the new
coordinates.

23. New coordinates of the triangle after rotation
= A (0, 0), B(0, 1), C(-1, 1)

24. Expansions and Compressions
(Dilations and Contractions)
Given a scalar r, define
22
: RRT  by T(x) = rx.
T is called a Contraction if 0 < r < 1, and a Dilation
if r > 1.

25. Projections

26. Projection Transformations

30. Geometry of Linear Transformation
• https://math.hmc.edu/calculus/hmc-mathematics-calculus-online-
tutorials/linear-algebra/geometry-of-linear-transformations/

31. • In this section we learn to understand matrices geometrically as functions,
or transformations. We briefly discuss transformations in general, then specialize to
matrix transformations, which are transformations that come from matrices.
• https://textbooks.math.gatech.edu/ila/matrix-transformations.html#matrix-trans-
matrices-functions
• https://www.youtube.com/watch?reload=9&v=kWW6fXV3OKk
• LT- Computer Graphics