This document defines and discusses linear transformations. It begins by defining a linear transformation as a function where the transformation of the sum of two vectors is equal to the sum of the individual transformations and where transformation and scalar multiplication commute. It then provides examples of determining if specific transformations are linear and evaluating linear transformations. Finally, it discusses properties of linear transformations such as being one-to-one, kernels, ranges, and whether a transformation is onto.

1. LINEAR TRANSFORMATION
MATH 15 - Linear Algebra
Department of Mathematics

2. At the end of this lesson, the students are
expected to:
•Define linear transformation.
•Determine if a transformation is linear transformation.
•Perform the transformation of given functions.
OBJECTIVES

3. DEFINITION:
A transformation 𝐿 is a linear transformation if
the following expressions hold:
For any real number:
Linear Transformation
]
[
]
[
]
[ v
L
u
L
v
u
L 


]
[
]
[ u
kL
ku
L 

4. EXAMPLE:
Determine if the following transformations are
linear transformation:
1. R2  R 3 :
2. R3  P 2 :
3. R3  M22:
LINEAR TRANSFORMATION


















y
y
x
x
y
x
L
4
3
2
2
z
yt
xt
z
y
x
T 












2





















z
y
z
y
y
x
y
x
z
y
x
K

5. EXAMPLE:
Determine if the following transformations are
linear transformation:
4. P3  P2 :
5. R3  R 2 :
6. R2  R:
LINEAR TRANSFORMATION
  y
xt
wt
z
yt
xt
wt
D 




 2
3 2
2
3
z
yt
xt
x
x
z
y
x
T 






 











2
2
3
1
 
3
y
x
y
x
K 








6. The following operations are tested to be linear
transformations. Evaluate the following linear
transformations:
1. P3  P2 :
Find D(5t3+2t2-3t+8)
2.R3  R 2
Find:
EVALUATION OF LINEAR TRANSFORMATION
  y
xt
wt
z
yt
xt
wt
D 




 2
3 2
2
3



















z
y
y
x
z
y
x
L
2




















8
4
1
.
3
6
4
. L
b
L
a

7. For the following items:
1. Given the transformation R2  R3 :
Find:
EVALUATION OF LINEAR TRANSFORMATION

















5
1
3
3
2
L

















1
2
6
8
6
L






9
6
L 





4
3
L 





11
8
L 





7
5
L

8. For the following items:
1. Given the transformation R3  R2 :
Find:
Note:
EVALUATION OF LINEAR TRANSFORMATION
  






3
2
i
L   






2
7
k
L
  






8
6
j
L










1
3
6
L











5
3
9
L











0
0
1
i











1
0
0
k











0
1
0
j

9. • Recall that a function is 1-1 if f(x) = f(y)
implies that x = y
Since a linear transformation is defined as a
function, the definition of 1-1 carries over to
linear transformations. That is
• Definition
A linear transformation L is 1-1 if for all
vectors u and v, L(u) = L(v)
implies that u = v
ONE TO ONE LINEAR TRANSFORMATION

10. DEFINITION:
• The kernel of a linear transformation L is the
set of all vectors v such that
L(v) = 0
Example: Find the kernel of
KERNEL

11. • The kernel of a linear transformation from a
vector space V to a vector space W is a
subspace of V.
THEOREM

12. • A linear transformation L is 1-1 if and only
if Ker(L) = 0.
THEOREM

13. Definition
Let L be a linear transformation from a vector
space V to a vector space W. Then
the range of L is the set of all
vectors w in W such that there is a v in V with
L(v) = w
Theorem
The range of a linear transformation
L from V to W is a subspace of W.
RANGE

14. Let L be the linear transformation from R2  R3 be
defined by
L(v) = Av
with
• A. Find a basis for Ker(L).
• B. Determine of L is 1-1.
• C. Find a basis for the range of L.
• D. Determine if L is onto.
EXAMPLE

15. TEXTBOOKS
Elementary Linear Algebra, Bernard Kolman and David
R. Hill, 7th ed., 2003
WEBSITES:
http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_a
nd_eigenspace
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