1. Lecture 14 Linear Transformation
Last Time
Applications of Linear Transformations
Introduction to Linear Transformations
The Kernel and Range of a Linear Transformation
Elementary Linear Algebra
R. Larsen et al. (6th Edition) TKUEE翁慶昌-NTUEE SCC_01_2009

2. Lecture 14: Linear Transformation
Today
The Kernel and Range of a Linear Transformation (Cont.)
Matrices for Linear Transformations
Transition Matrix and Similarity
Eigenvalues and Eigenvectors
Diagonalization
Reading Assignment: Secs 6.3 – 7.2
Next Time
Final Exam
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3. The Kernel and Range of a Linear Transformation (Cont.)
Today
The Kernel and Range of a Linear Transformation (Cont.)
Matrices for Linear Transformations
Transition Matrix and Similarity
Eigenvalues and Eigenvectors
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4. 6.2 The Kernel and Range of a Linear Transformation
Kernel of a linear transformation T:
Let be a linear transformation
Then the set of all vectors v in V that satisfy is called the kernel of T and is denoted by ker(T).
Ex 1: (Finding the kernel of a linear transformation)
Sol:
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5. Range of a linear transformation T:
Corollary to Thm 6.3:
Range of a linear transformation T:
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6. Notes:
Corollary to Thm 6.4:
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7. Rank of a linear transformation T:V→W:
Nullity of a linear transformation T:V→W:
Note:
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8. Thm 6.5: (Sum of rank and nullity)
Pf:
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9. One-to-one:
one-to-one
not one-to-one
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10. (T is onto W when W is equal to the range of T.)

11. Thm 6.6: (One-to-one linear transformation)
Pf:

12. Ex 10: (One-to-one and not one-to-one linear transformation)

13. Thm 6.7: (Onto linear transformation)
Thm 6.8: (One-to-one and onto linear transformation)
Pf:

14. Ex 11: Sol: T:Rn→Rm dim(domain of T) rank(T) nullity(T) 1-1 onto
(a)T:R3→R3 3
Yes
(b)T:R2→R3 2 No
(c)T:R3→R2 1
(d)T:R3→R3

15. Thm 6.9: (Isomorphic spaces and dimension)
Isomorphism:
Thm 6.9: (Isomorphic spaces and dimension)
Pf:
Two finite-dimensional vector space V and W are isomorphic if and only if they are of the same dimension.

16. It can be shown that this L.T. is both 1-1 and onto.
Thus V and W are isomorphic.

17. Ex 12: (Isomorphic vector spaces)
The following vector spaces are isomorphic to each other.

18. Keywords in Section 6.2:
kernel of a linear transformation T: 線性轉換T的核空間
range of a linear transformation T: 線性轉換T的值域
rank of a linear transformation T: 線性轉換T的秩
nullity of a linear transformation T: 線性轉換T的核次數
one-to-one: 一對一
onto: 映成
isomorphism(one-to-one and onto): 同構
isomorphic space: 同構的空間

19. The Kernel and Range of a Linear Transformation (Cont.)
Today
The Kernel and Range of a Linear Transformation (Cont.)
Matrices for Linear Transformations
Transition Matrix and Similarity
Eigenvalues and Eigenvectors

20. 6.3 Matrices for Linear Transformations
Two representations of the linear transformation T:R3→R3 :
Three reasons for matrix representation of a linear transformation:
It is simpler to write.
It is simpler to read.
It is more easily adapted for computer use.

21. Thm 6.10: (Standard matrix for a linear transformation)

22. Pf:

24. Ex 1: (Finding the standard matrix of a linear transformation)
Sol:
Vector Notation Matrix Notation

25. Check:
Note:

26. Composition of T1:Rn→Rm with T2:Rm→Rp :
Thm 6.11: (Composition of linear transformations)

27. Pf:
Note:

28. Ex 3: (The standard matrix of a composition)
Sol:

30. Inverse linear transformation:
Note:
If the transformation T is invertible, then the inverse is unique and denoted by T–1 .

31. Thm 6.12: (Existence of an inverse transformation)
T is invertible.
T is an isomorphism.
A is invertible.
Note:
If T is invertible with standard matrix A, then the standard matrix for T–1 is A–1 .

32. Ex 4: (Finding the inverse of a linear transformation)
Show that T is invertible, and find its inverse.
Sol:

34. the matrix of T relative to the bases B and B':
Thus, the matrix of T relative to the bases B and B' is

35. Transformation matrix for nonstandard bases:

37. Ex 5: (Finding a matrix relative to nonstandard bases)
Sol:

38. Ex 6:
Sol:
Check:

39. Notes:

40. Keywords in Section 6.3: standard matrix for T: T 的標準矩陣
composition of linear transformations: 線性轉換的合成
inverse linear transformation: 反線性轉換
matrix of T relative to the bases B and B' : T對應於基底B到B'的矩陣
matrix of T relative to the basis B: T對應於基底B的矩陣

41. 6.4 Transition Matrices and Similarity

42. Two ways to get from to :

43. Ex 2: (Finding a matrix for a linear transformation)
Sol:

44. Ex 3: (Finding a matrix for a linear transformation)
Sol:

45. Thm 6.13: (Properties of similar matrices)
Similar matrix:
For square matrices A and A‘ of order n, A‘ is said to be similar to A if there exist an invertible matrix P s.t.
Thm 6.13: (Properties of similar matrices)
Let A, B, and C be square matrices of order n.
Then the following properties are true.
(1) A is similar to A.
(2) If A is similar to B, then B is similar to A.
(3) If A is similar to B and B is similar to C, then A is similar to C.
Pf:

46. Ex 4: (Similar matrices)

47. Ex 5: (A comparison of two matrices for a linear transformation)
Sol:

49. Notes: Computational advantages of diagonal matrices:

50. 7.1 Eigenvalues and Eigenvectors
Eigenvalue problem:
If A is an nn matrix, do there exist nonzero vectors x in Rn such that Ax is a scalar multiple of x？
Eigenvalue and eigenvector:
A：an nn matrix
：a scalar
x： a nonzero vector in Rn
Geometrical Interpretation
Eigenvalue
Eigenvector

51. Lecture 14: Linear Transformation
Today
Matrices for Linear Transformations
Transition Matrix and Similarity
Eigenvalues and Eigenvectors
Reading Assignment: Secs
Scope:
Sections : 70%,
Sections 1.1 – 4.5: 30%
Tip: Practice your homework problems and really understand
Makeup Lecture
Diagonalization
Symmetric Matrices and Orthogonal Diagonalization
Applications
1/19/2009 2:20 – 5:10 Mgmt 101
Reading Assignment: Secs
14- 51