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

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 14 - 2

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 14 - 3

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 - 4

Range of a linear transformation T: Corollary to Thm 6.3: Range of a linear transformation T: 14 - 5

Notes: Corollary to Thm 6.4: 14 - 6

Rank of a linear transformation T:V→W: Nullity of a linear transformation T:V→W: Note: 14 - 7

Thm 6.5: (Sum of rank and nullity) Pf: 14 - 8

One-to-one: one-to-one not one-to-one 14 - 9

(T is onto W when W is equal to the range of T.) 14 - 10

Thm 6.6: (One-to-one linear transformation) Pf: 14 - 11

Ex 10: (One-to-one and not one-to-one linear transformation) 14 - 12

Thm 6.7: (Onto linear transformation) Thm 6.8: (One-to-one and onto linear transformation) Pf: 14 - 13

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 14 - 14

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. 14 - 15

It can be shown that this L.T. is both 1-1 and onto. Thus V and W are isomorphic. 14 - 16

Ex 12: (Isomorphic vector spaces) The following vector spaces are isomorphic to each other. 14 - 17

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 14 - 18

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 14 - 19

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. 14 - 20

Thm 6.10: (Standard matrix for a linear transformation) 14 - 21

Pf: 14 - 22

14 - 23

Ex 1: (Finding the standard matrix of a linear transformation) Sol: Vector Notation Matrix Notation 14 - 24

Check: Note: 14 - 25

Composition of T1:Rn→Rm with T2:Rm→Rp : Thm 6.11: (Composition of linear transformations) 14 - 26

Pf: Note: 14 - 27

Ex 3: (The standard matrix of a composition) Sol: 14 - 28

14 - 29

Inverse linear transformation: Note: If the transformation T is invertible, then the inverse is unique and denoted by T–1 . 14 - 30

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 . 14 - 31

Ex 4: (Finding the inverse of a linear transformation) Show that T is invertible, and find its inverse. Sol: 14 - 32

14 - 33

the matrix of T relative to the bases B and B': Thus, the matrix of T relative to the bases B and B' is 14 - 34

Transformation matrix for nonstandard bases: 14 - 35

14 - 36

Ex 5: (Finding a matrix relative to nonstandard bases) Sol: 14 - 37

Ex 6: Sol: Check: 14 - 38

Notes: 14 - 39

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 14 - 40

6.4 Transition Matrices and Similarity 14 - 41

Two ways to get from to : 14 - 42

Ex 2: (Finding a matrix for a linear transformation) Sol: 14 - 43

Ex 3: (Finding a matrix for a linear transformation) Sol: 14 - 44

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: 14 - 45

Ex 4: (Similar matrices) 14 - 46

Ex 5: (A comparison of two matrices for a linear transformation) Sol: 14 - 47

14 - 48

Notes: Computational advantages of diagonal matrices: 14 - 49

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 14 - 50

Lecture 14: Linear Transformation Today Matrices for Linear Transformations Transition Matrix and Similarity Eigenvalues and Eigenvectors Reading Assignment: Secs 6.3-7.1 Scope: Sections 4.6-6.5: 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 7.1-7.2 14- 51