CHAPTER 6 LINEAR TRANSFORMATIONS 6.1 Introduction to Linear Transformations 6.2 The Kernel and Range of a Linear Transformation 6.3 Matrices for Linear Transformations 6.4 Transition Matrices and Similarity 6.5 Applications of Linear Transformations Elementary Linear Algebra 投影片設計編製者 R. Larson (7 Edition) 淡江大學 電機系 翁慶昌 教授

1/90 CH 6 Linear Algebra Applied Multivariate Statistics (p.298) Circuit Design (p.316) Control Systems (p.308) Weather (p.325) Computer Graphics (p.332)

2/ Introduction to Linear Transformations Function T that maps a vector space V into a vector space W: V: the domain of T W: the codomain of T Elementary Linear Algebra: Section 6.1, p.292

3/90 Image of v under T: If v is in V and w is in W such that Then w is called the image of v under T. the range of T: The set of all images of vectors in V. the preimage of w: The set of all v in V such that T(v)=w. Elementary Linear Algebra: Section 6.1, p.292

4/90 Ex 1: (A function from R 2 into R 2 ) (a) Find the image of v=(-1,2). (b) Find the preimage of w=(-1,11) Sol: Thus {(3, 4)} is the preimage of w=(-1, 11). Elementary Linear Algebra: Section 6.1, p.292

5/90 Linear Transformation (L.T.): Elementary Linear Algebra: Section 6.1, p.293

6/90 Notes: (1) A linear transformation is said to be operation preserving. Addition in V Addition in W Scalar multiplication in V Scalar multiplication in W (2) A linear transformation from a vector space into itself is called a linear operator. Elementary Linear Algebra: Section 6.1, p.293

7/90 Ex 2: (Verifying a linear transformation T from R 2 into R 2 ) Pf: Elementary Linear Algebra: Section 6.1, p.293

8/90 Therefore, T is a linear transformation. Elementary Linear Algebra: Section 6.1, p.293

9/90 Ex 3: (Functions that are not linear transformations) Elementary Linear Algebra: Section 6.1, p.294

10/90 Notes: Two uses of the term “linear”. (1) is called a linear function because its graph is a line. (2) is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication. Elementary Linear Algebra: Section 6.1, p.294

11/90 Zero transformation: Identity transformation: Thm 6.1: (Properties of linear transformations) Elementary Linear Algebra: Section 6.1, p.294

12/90 Ex 4: (Linear transformations and bases) Let be a linear transformation such that Sol: (T is a L.T.) Find T(2, 3, -2). Elementary Linear Algebra: Section 6.1, p.295

13/90 Ex 5: (A linear transformation defined by a matrix) The function is defined as Sol: (vector addition) (scalar multiplication) Elementary Linear Algebra: Section 6.1, p.295

14/90 Thm 6.2: (The linear transformation given by a matrix) Let A be an m  n matrix. The function T defined by is a linear transformation from R n into R m. Note: Elementary Linear Algebra: Section 6.1, p.296

15/90 Show that the L.T. given by the matrix has the property that it rotates every vector in R 2 counterclockwise about the origin through the angle . Ex 7: (Rotation in the plane) Sol: (polar coordinates) r ： the length of v  ： the angle from the positive x-axis counterclockwise to the vector v Elementary Linear Algebra: Section 6.1, p.297

16/90 r ： the length of T(v)  +  ： the angle from the positive x-axis counterclockwise to the vector T(v) Thus, T(v) is the vector that results from rotating the vector v counterclockwise through the angle . Elementary Linear Algebra: Section 6.1, p.297

17/90 is called a projection in R 3. Ex 8: (A projection in R 3 ) The linear transformation is given by Elementary Linear Algebra: Section 6.1, p.298

18/90 Show that T is a linear transformation. Ex 9: (A linear transformation from M m  n into M n  m ) Sol: Therefore, T is a linear transformation from M m  n into M n  m. Elementary Linear Algebra: Section 6.1, p.298

19/90 Key Learning in Section 6.1 ▪Find the image and preimage of a function. ▪Show that a function is a linear transformation, and find a linear transformation.

20/90 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

21/ 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: Elementary Linear Algebra: Section 6.2, p.303

22/90 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: Elementary Linear Algebra: Section 6.2, p.303

23/90 Ex 5: (Finding the kernel of a linear transformation) Sol: Elementary Linear Algebra: Section 6.2, p.304

24/90 Elementary Linear Algebra: Section 6.2, p.304

25/90 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. Elementary Linear Algebra: Section 6.2, p.305

26/90 Ex 6: (Finding a basis for the kernel) Find a basis for ker(T) as a subspace of R 5. Elementary Linear Algebra: Section 6.2, p.30 5

27/90 Sol: Elementary Linear Algebra: Section 6.2, p.305

28/90 Corollary to Thm 6.3: Range of a linear transformation T: Elementary Linear Algebra: Section 6.2, p.305

29/90 Thm 6.4: (The range of T is a subspace of W) Pf: Elementary Linear Algebra: Section 6.2, p.306

30/90 Notes: Corollary to Thm 6.4: Elementary Linear Algebra: Section 6.2, p.306

31/90 Ex 7: (Finding a basis for the range of a linear transformation) Find a basis for the range of T. Elementary Linear Algebra: Section 6.2, p.307

32/90 Sol: Elementary Linear Algebra: Section 6.2, p.307

33/90 Rank of a linear transformation T:V→W: Nullity of a linear transformation T:V→W: Note: Elementary Linear Algebra: Section 6.2, p.307

34/90 Thm 6.5: (Sum of rank and nullity) Pf: Elementary Linear Algebra: Section 6.2, p.307

35/90 Ex 8: (Finding the rank and nullity of a linear transformation) Sol: Elementary Linear Algebra: Section 6.2, p.308

36/90 Ex 9: (Finding the rank and nullity of a linear transformation) Sol: Elementary Linear Algebra: Section 6.2, p.308

37/90 One-to-one: one-to-onenot one-to-one Elementary Linear Algebra: Section 6.2, p.309

38/90 Onto: (T is onto W when W is equal to the range of T.) Elementary Linear Algebra: Section 6.2, p.309

39/90 Thm 6.6: (One-to-one linear transformation) Pf: Elementary Linear Algebra: Section 6.2, p.309

40/90 Ex 10: (One-to-one and not one-to-one linear transformation) Elementary Linear Algebra: Section 6.2, p.309

41/90 Thm 6.7: (Onto linear transformation) Thm 6.8: (One-to-one and onto linear transformation) Pf: Elementary Linear Algebra: Section 6.2, p.310

42/90 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 Elementary Linear Algebra: Section 6.2, p.310

43/90 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. Elementary Linear Algebra: Section 6.2, p.311

44/90 It can be shown that this L.T. is both 1-1 and onto. Thus V and W are isomorphic. Elementary Linear Algebra: Section 6.2, p.311

45/90 Ex 12: (Isomorphic vector spaces) The following vector spaces are isomorphic to each other. Elementary Linear Algebra: Section 6.2, p.311

46/90 Key Learning in Section 6.2 ▪Find the kernel of a linear transformation. ▪Find a basis for the range, the rank, and the nullity of a linear transformation. ▪Determine whether a linear transformation is one-to-one or onto. ▪Determine whether vector spaces are isomorphic.

47/90 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

48/ Matrices for Linear Transformations 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. Two representations of the linear transformation T:R 3 →R 3 : Elementary Linear Algebra: Section 6.3, p.314

49/90 Thm 6.10: (Standard matrix for a linear transformation) Elementary Linear Algebra: Section 6.3, p.314

50/90 Pf: Elementary Linear Algebra: Section 6.3, p.315

51/90 Elementary Linear Algebra: Section 6.3, p.315

52/90 Ex 1: (Finding the standard matrix of a linear transformation) Sol: Vector Notation Matrix Notation Elementary Linear Algebra: Section 6.3, p.315

53/90 Note: Check: Elementary Linear Algebra: Section 6.3, p.315

54/90 Ex 2: (Finding the standard matrix of a linear transformation) Sol: Notes: (1) The standard matrix for the zero transformation from R n into R m is the m  n zero matrix. (2) The standard matrix for the zero transformation from R n into R n is the n  n identity matrix I n Elementary Linear Algebra: Section 6.3, p.316

55/90 Composition of T 1 :R n →R m with T 2 :R m →R p : Thm 6.11: (Composition of linear transformations) Elementary Linear Algebra: Section 6.3, p.317

56/90 Pf: Note: Elementary Linear Algebra: Section 6.3, p.317

57/90 Ex 3: (The standard matrix of a composition) Sol: Elementary Linear Algebra: Section 6.3, p.318

58/90 Elementary Linear Algebra: Section 6.3, p.318

59/90 Inverse linear transformation: Note: If the transformation T is invertible, then the inverse is unique and denoted by T –1. Elementary Linear Algebra: Section 6.3, p.318

60/90 Thm 6.12: (Existence of an inverse transformation) Note: If T is invertible with standard matrix A, then the standard matrix for T –1 is A –1. (1)T is invertible. (2)T is an isomorphism. (3)A is invertible. Elementary Linear Algebra: Section 6.3, p.319

61/90 Ex 4: (Finding the inverse of a linear transformation) Sol: Show that T is invertible, and find its inverse. Elementary Linear Algebra: Section 6.3, p.319

62/90 Elementary Linear Algebra: Section 6.3, p.319

63/90 the matrix of T relative to the bases B and B': Thus, the matrix of T relative to the bases B and B' is Elementary Linear Algebra: Section 6.3, p.320

64/90 Transformation matrix for nonstandard bases: Elementary Linear Algebra: Section 6.3, p.320

65/90 Elementary Linear Algebra: Section 6.3, p.320

66/90 Ex 5: (Finding a matrix relative to nonstandard bases) Sol: Elementary Linear Algebra: Section 6.3, p.320

67/90 Ex 6: Sol: Check: Elementary Linear Algebra: Section 6.3, p.321

68/90 Notes: Elementary Linear Algebra, Section 6.3, p.321

69/90 Key Learning in Section 6.3 ▪Find the standard matrix for a linear transformation. ▪Find the standard matrix for the composition of linear transformations and find the inverse of an invertible linear transformation. ▪Find the matrix for a linear transformation relative to a nonstandard basis.

70/90 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

71/ Transition Matrices and Similarity Elementary Linear Algebra: Section 6.4, p.324

72/90 Two ways to get from to : Elementary Linear Algebra: Section 6.4, p.324

73/90 Ex 1: (Finding a matrix for a linear transformation) Sol: Elementary Linear Algebra: Section 6.4, p.324

74/90 Elementary Linear Algebra: Section 6.4, p.324

75/90 Ex 2: (Finding a matrix for a linear transformation) Sol: Elementary Linear Algebra: Section 6.4, p.325

76/90 Ex 3: (Finding a matrix for a linear transformation) Sol: Elementary Linear Algebra: Section 6.4, p.325

77/90 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: Elementary Linear Algebra: Section 6.4, p.326

78/90 Ex 4: (Similar matrices) Elementary Linear Algebra: Section 6.4, p.326

79/90 Ex 5: (A comparison of two matrices for a linear transformation) Sol: Elementary Linear Algebra: Section 6.4, p.327

80/90 Elementary Linear Algebra: Section 6.4, p.327

81/90 Notes: Computational advantages of diagonal matrices: Elementary Linear Algebra: Section 6.4, p.327

82/90 Key Learning in Section 6.4 ▪Find and use a matrix for a linear transformation. ▪Show that two matrices are similar and use the properties of similar matrices.

83/90 Keywords in Section 6.4 matrix of T relative to B: T 相對於 B 的矩陣 matrix of T relative to B' : T 相對於 B' 的矩陣 transition matrix from B' to B : 從 B' 到 B 的轉移矩陣 transition matrix from B to B' : 從 B 到 B' 的轉移矩陣 similar matrix: 相似矩陣

84/90 Key Learning in Section 6.5 ▪Identify linear transformations defined by reflections, expansions, ▪contractions, or shears in R 2. ▪Use a linear transformation to rotate a figure in R 3.

85/90 Multivariate Statistics Many multivariate statistical methods can use linear transformations. For instance, in a multiple regression analysis, there are two or more independent variables and a single dependent variable. A linear transformation is useful for finding weights to be assigned to the independent variables to predict the value of the dependent variable. Also, in a canonical correlation analysis, there are two or more independent variables and two or more dependent variables. Linear transformations can help find a linear combination of the independent variables to predict the value of a linear combination of the dependent variables. 6.1 Linear Algebra Applied Elementary Linear Algebra: Section 6.1, p.298

86/90 Control Systems A control system, such as the one shown for a dairy factory, processes an input signal x k and produces an output signal x k+1. Without external feedback, the difference equation x k+1 = Ax k a linear transformation where x i is an n  1 vector and A is an n  n matrix, can model the relationship between the input and output signals. Typically, however, a control system has external feedback, so the relationship becomes x k+1 = Ax k +Bu k where B is an n  m matrix and u k is an m  1 input, or control, vector. A system is called controllable when it can reach any desired final state from its initial state in or fewer steps. If A and B make up a controllable system, then the rank of the controllability matrix [B AB A 2 B... A n-1 B] is equal to n. 6.2 Linear Algebra Applied Elementary Linear Algebra: Section 6.2, 308

87/90 Circuit Design Ladder networks are useful tools for electrical engineers involved in circuit design. In a ladder network, the output voltage and current of one circuit are the input voltage and current of the circuit next to it. In the ladder network shown below, linear transformations can relate the input and output of an individual circuit (enclosed in a dashed box). Using Kirchhoff’s Voltage and Current Laws and Ohm’s Law, and 6.3 Linear Algebra Applied Elementary Linear Algebra: Section 6.3, p.316

88/90 Circuit Design A composition can relate the input and output of the entire ladder network, that is, and to and Discussion on the composition of linear transformations begins on the following page. 6.3 Linear Algebra Applied Elementary Linear Algebra: Section 6.3, p.316

89/90 Weather In Section 2.5, you studied stochastic matrices and state matrices. A Markov chain is a sequence {x n }of state matrices that are probability vectors related by the linear transformation x k+1 = Px k,where P, the transition matrix from one state to the next, is a stochastic matrix [p ij ]. For instance, suppose that it has been established, through studying extensive weather records, that the probability p 21 of a stormy day following a sunny day is 0.1 and the probability p 22 of a stormy day following a stormy day is 0.2. The transition matrix can be written as 6.4 Linear Algebra Applied Elementary Linear Algebra: Section 6.4, p.325

90/90 Computer Graphics The use of computer graphics is common in many fields. By using graphics software, a designer can “see” an object before it is physically created. Linear transformations can be useful in computer graphics. To illustrate with a simplified example, only 23 points in R 3 make up the images of the toy boat shown in the figure at the left. Most graphics software can use such minimal information to generate views of an image from any perspective, as well as color, shade, and render as appropriate. Linear transformations, specifically those that produce rotations in R 3 can represent the different views. The remainder of this section discusses rotation in R Linear Algebra Applied Elementary Linear Algebra: Section 6.5, p.332