The document discusses one-to-one and onto linear transformations. It states that a transformation T is one-to-one if each b in the codomain is the image of at most one x in the domain. T is one-to-one if the columns of its matrix A are linearly independent. T is onto if each b in the codomain is the image of at least one x in the domain, meaning the columns of A span the codomain. Examples are provided to illustrate one-to-one and onto transformations along with references to video explanations.

1. One to One and Onto
Transformations
Dr. Farhana Shaheen

2. Linear Transformations Representations

3. Theorem 4: Topic 1.4
• Let A be an mxn matrix. Then the following statements are logically
equivalent:
• 1. For each b in , the equation Ax = b has a solution.
• 2. Each b in is a linear combination of the columns of A
• 3. The columns of A span .
• 4. A has a pivot position in every row.
m
R
m
R
m
R

4. Functions: Domain, Codomain and Range

5. Linear Transformation
Let V be an n-dimensional vector space, let W be an m-dimensional
vector space, and let
T be any linear transformation from V to W . To associate a matrix
with T, choose (ordered) bases B and C for V and W, respectively.
Given any x in V , the coordinate vector [x]B is in ,
and the coordinate vector of its image, [T(x)]C , is in , as shown in
Figure 1:
m
R
𝑅 𝑛

6. Linear Transformation

7. One-to-One Transformation
• A mapping is said to be One-to-one if each b in is the
image of at most one x in .
• So, T is One-to-one if for each b in , the equation T(x)=b (Ax = b) has
either a unique solution or none at all.
• That is, “Is T 1-to-1?” is a Uniqueness question.
• Note 1: The mapping T is not one-to-one when some b in is the image
of more than one vector in .
• Note 2: Not One –to-one mean Consistent (with Infinite solutions).
mn
RRT :
mn
RRT :
m
R
n
R
m
R
m
Rn
R

8. One-to-one
Transformation
• A mapping
is said to be One-to-
one if each b in
is the image of at most
one x in .
mn
RRT :
m
R
𝑅 𝑛

9. Condition for One –to- one Transformation
Let be a linear transformation, and let A be the standard
Matrix for T. Then:
1. T is One-to –one if and only if the columns of A are Linearly Independent.
2. T is one-to-one if and only if T(x) = 0 or Ax = 0 has only the trivial solution.
3. T is one-to-one if and only if there is a pivot in every column of A in RREF.
mn
RRT :

10. Result: (page:77)
• Let be a Linear transformation. Then T is One-to-one if and
only if the equation T(x) = 0 has only the Trivial Solution.
• (Reason: As T is linear, T(0) = 0. T is one-to-one means, it has at most
one solution, which is trivial solution).
mn
RRT :

11. • https://www.youtube.com/watch?v=53KDr8y7VnQ

12. Onto Transformation
• A mapping is said to be Onto if each b in is the
image of at least one x in .
• So, T is Onto if the range of T all of the Codomain .
• That is, for each b in the Codomain, there exist at least one solution of
T(x) = b.
• Note 1: Onto implies Existence of Solution– (Consistent)
• Note 2: Not Onto means No Solution---- (Inconsistent)
mn
RRT :
mn
RRT :
m
R
m
R
n
R
m
R m
R

13. Onto Transformation

14. Condition for Onto Transformation
Let be a linear transformation, and let A be the standard
Matrix for T. Then:
T maps Onto if and only if the columns of A Span ;
(if and only if for each b in , the equation Ax= b is consistent)
(if and only if for each b in , the equation T(x) = b has at least one
solution. Every vector in is a linear combination of the columns of A)
mn
RRT :
m
R
m
R
n
R m
R
m
R
m
R

15. • https://www.youtube.com/watch?v=-M0kHX-rIv4

16. Ex:1.9 Qs: 17 page 79
• Show that T is a linear transformation by finding a matrix A that implements
the mapping.
• Find:

18. Q4U: One-to-One and Onto Transformation
• Note: T is one- to one if the standard
matrix has Linearly Independent
Columns. For
• Find:
• https://www.youtube.com/watch?v=0
G-edqxScZQ

19. Quiz-1

20. One-to-one and Onto Transformation
• Note: 1 The Reflection Transformations are all One-to-one and
• Map onto .
• Note: 2 The Projection transformations are not one-to-one and also do not
map onto . T(0) = 0 and T ( ) = 0.

2e
2
R2
R
2
R2
R






00
01






10
01

21. • The Reflection
Transformations are all
One-to-one and
• Map onto .
2
R 2
R

23. Example:
• Let T be the Linear transformation whose standard matrix is
(i) Does T maps R4 onto R3? (Existence)
Note: (A has pivot position in each row. So Ax=b is consistent)
(i) Is T a one-to-one mapping? (Uniqueness) (Three basic and one free
variable. So infinite solutions)













5000
3120
1841
A

24. Dr. Farhana Shaheen
Linear Transformations Examples- Please Watch
https://www.youtube.com/watch?v=ztjvnzjejwk
Dr. Farhana Shaheen
https://www.youtube.com/watch?v=J2bjzpyW6ro
Dr. Farhana Shaheen
https://www.youtube.com/watch?v=3HB16IOEZRc
Dr. Farhana Shaheen
https://www.youtube.com/watch?v=-M0kHX-rIv4 Onto LT
https://www.youtube.com/watch?v=53KDr8y7VnQ One-to-one LT