1. Lecture 14 Linear Transformation Last Time - Mathematical Models and Least Square Analysis - Inner Product Space Applications - Introduction to Linear Transformations Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE 翁慶昌 -NTUEE SCC_01_2008

2. 14- 2 Lecture 14: Linear Transformation Today The Kernel and Range of a Linear Transformation Matrices for Linear Transformations Transition Matrix and Similarity Reading Assignment: Secs 6.2-6.4 Next Time Applications of Linear Transformations Eigenvalues and Eigenvectors Diagonalization Reading Assignment: Secs 6.5-7.2

3. What Have You Actually Learned about LSPs So Far? 14- 3

4. Keywords in Section 6.1: function: 函數 domain: 論域 codomain: 對應論域 image of v under T: 在 T 映射下 v 的像 range of T: T 的值域 preimage of w: w 的反像 linear transformation: 線性轉換 linear operator: 線性運算子 zero transformation: 零轉換 identity transformation: 相等轉換 14 - 4

5. Today The Kernel and Range of a Linear Transformation Matrices for Linear Transformations Transition Matrix and Similarity 14 - 5

6. 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: 14 - 6

7. 14 - 7 Ex 2: (The kernel of the zero and identity transformations) (a) T(v)=0 (the zero transformation ) (b) T(v)=v (the identity transformation ) Ex 3: (Finding the kernel of a linear transformation) Sol:

8. 14 - 8 Ex 5: (Finding the kernel of a linear transformation) Sol:

9. 14 - 9

10. Thm 6.3: (The kernel is a subspace of V) The kernel of a linear transformation is a subspace of the domain V. Pf: Note: The kernel of T is sometimes called the nullspace of T. 14 - 10

11. 14 - 11 Ex 6: (Finding a basis for the kernel) Find a basis for ker(T) as a subspace of R5.

12. 14 - 12 Sol:

13. Corollary to Thm 6.3: Range of a linear transformation T: 14 - 13

14. 14 - 14 Thm 6.4: (The range of T is a subspace of W) Pf:

15. 14 - 15 Notes: Corollary to Thm 6.4:

16. 14 - 16 Ex 7: (Finding a basis for the range of a linear transformation) Find a basis for the range of T.

17. 14 - 17 Sol:

18. 14 - 18 Rank of a linear transformation T:V → W: Nullity of a linear transformation T:V → W: Note:

19. 14 - 19 Thm 6.5: (Sum of rank and nullity) Pf:

20. 6 - 20 Ex 8: (Finding the rank and nullity of a linear transformation) Sol:

21. 14 - 21 Ex 9: (Finding the rank and nullity of a linear transformation) Sol:

22. 14 - 22 One-to-one: one-to-onenot one-to-one

23. 14 - 23 Onto: (T is onto W when W is equal to the range of T.)

24. 14 - 24 Thm 6.6: (One-to-one linear transformation) Pf:

25. 14 - 25 Ex 10: (One-to-one and not one-to-one linear transformation)

26. 14 - 26 Thm 6.7: (Onto linear transformation) Thm 6.8: (One-to-one and onto linear transformation) Pf:

27. 14 - 27 Ex 11: Sol: T:Rn→RmT:Rn→Rm dim(domain of T) rank(T)nullity(T)1-1onto (a)T:R3→R3(a)T:R3→R3 330Yes (b)T:R2→R3(b)T:R2→R3 220 No (c)T:R3→R2(c)T:R3→R2 321 Yes (d)T:R3→R3(d)T:R3→R3 321No

28. 14 - 28 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.

29. 14 - 29 It can be shown that this L.T. is both 1-1 and onto. Thus V and W are isomorphic.

30. 14 - 30 Ex 12: (Isomorphic vector spaces) The following vector spaces are isomorphic to each other.

32. 14 - 32 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