Presentation is loading. Please wait.

Presentation is loading. Please wait.

Chap. 6 Linear Transformations

Similar presentations


Presentation on theme: "Chap. 6 Linear Transformations"— Presentation transcript:

1 Chap. 6 Linear Transformations
Linear Algebra Ming-Feng Yeh Department of Electrical Engineering Lunghwa University of Science and Technology

2 6.1 Introduction to Linear Transformation
Learn about functions that map a vector space V into a vector space W --- T: V  W V: domain of T range v w image of v T: V  W W: codomain of T Ming-Feng Yeh Chapter 6

3 Section 6-1 Map If v is in V and w is in W s.t. T(v) = w, then w is called the image of v under T. The set of all images of vectors in V is called the range of T. The set of all v in V s.t. T(v) = w is called the preimage of w. Ming-Feng Yeh Chapter 6

4 Ex. 1: A function from R2 into R2
Section 6-1 Ex. 1: A function from R2 into R2 For any vector v = (v1, v2) in R2, and let T: R2 R2 be defined by T(v1, v2) = (v1  v2, v1 + 2v2) Find the image of v = (1, 2) T(1, 2) =(1 2, 1+2·2) = (3, 3) Find the preimage of w = (1, 11) T(v1, v2) = (v1  v2, v1 + 2v2) = (1, 11)  v1  v2 = 1; v1 + 2v2 = 11  v1 = 3; v2 = 4 Ming-Feng Yeh Chapter 6

5 Linear Transformation
Section 6-1 Linear Transformation Let V and W be vector spaces. The function T: V  W is called a linear transformation of V into W if the following two properties are true for all u and v in V and for any scalar c. 1. T(u + v) = T(u) + T(v) 2. T(cu) = cT(u) A linear transformation is said to be operation reserving (the operations of addition and scalar multiplication). Ming-Feng Yeh Chapter 6

6 Ex. 2: Verifying a linear transformation from R2 into R2
Section 6-1 Ex. 2: Verifying a linear transformation from R2 into R2 Show that the function T(v1, v2) = (v1  v2, v1 + 2v2) is a linear transformation from R2 into R2. Let v = (v1, v2) and u = (u1, u2) Vector addition: v + u = (v1 + u1, v2 + u2) T(v + u) = T(v1 + u1, v2 + u2) = ( (v1 + u1)  (v2 + u2), (v1 + u1) + 2(v2 + u2) ) = (v1  v2 , v1 + 2v2) + (u1  u2, u1 +2u2) = T(v) + T(u) Scalar multiplication: cv = c(v1, v2) = (cv1, cv2) T(cv) = (cv1  cv2, cv1+ 2cv2) = c(v1  v2, v1+ 2v2) = cT(v) Therefore T is a linear transformation. Ming-Feng Yeh Chapter 6

7 Ex. 3: Not linear transformation
Section 6-1 Ex. 3: Not linear transformation f(x) = sin(x) In general, sin(x1 + x2)  sin(x1) + sin(x2) f(x) = x2 In general, f(x) = x + 1 f(x1 + x2) = x1 + x2 + 1 f(x1) + f(x2) = (x1 + 1) + (x2 + 1) = x1 + x2 + 2 Thus, f(x1 + x2)  f(x1) + f(x2) Ming-Feng Yeh Chapter 6

8 Linear Operation & Zero / Identity Transformation
Section 6-1 Linear Operation & Zero / Identity Transformation A linear transformation T: V  V from a vector space into itself is called a linear operator. Zero transformation (T: V  W): T(v) = 0, for all v Identity transformation (T: V  V): T(v) = v, for all v Ming-Feng Yeh Chapter 6

9 Thm 6.1: Linear transformations
Section 6-1 Thm 6.1: Linear transformations Let T be a linear transformation from V into W, where u and v are in V. Then the following properties are true. 1. T(0) = 0 2. T(v) = T(v) 3. T(u  v) = T(u)  T(v) 4. If v = c1v1 + c2v2 + … + cnvn, then T(v) = c1T(v1) + c2T(v2) + … + cnT(vn) Ming-Feng Yeh Chapter 6

10 Section 6-1 Proof of Theorem 6.1 1. Note that 0v = 0. Then it follows that T(0) = T(0v) = 0T(v) = 0 2. Follow from v = (1)v, which implies that T(v) = T[(1)v] = (1)T(v) = T(v) 3. Follow from u  v = u + (v), which implies that T(u  v) = T[u + (1)v] = T(u) + (1)T(v) = T(u)  T(v) 4. Left to you Ming-Feng Yeh Chapter 6

11 Section 6-1 Remark of Theorem 6.1 A linear transformation T: V  W is determined completely by its action on a basis of V. If {v1, v2, …, vn} is a basis for the vector space V and if T(v1), T(v2), …, T(vn) are given, then T(v) is determined for any v in V. Ming-Feng Yeh Chapter 6

12 Ex 4: Linear transformations and bases
Section 6-1 Ex 4: Linear transformations and bases Let T: R3  R3 be a linear transformation s.t. T(1, 0, 0) = (2, 1, 4); T(0, 1, 0) = (1, 5, 2); T(0, 0, 1) = (0, 3, 1). Find T(2, 3, 2). (2, 3, 2) = 2(1, 0, 0) + 3(0, 1, 0) 2(0, 0, 1) T(2, 3, 2) = 2T(1, 0, 0) + 3T(0, 1, 0) 2T(0, 0, 1) = 2(2, 1, 4) + 3(1, 5, 2) 2(0, 3, 1) = (7, 7, 0) Ming-Feng Yeh Chapter 6

13 Ex 5: Linear transformation defined by a matrix
Section 6-1 Ex 5: Linear transformation defined by a matrix The function T: R2  R3 is defined as follows Find T(v), where v = (2, 1) Therefore, T(2, 1) = (6, 3, 0) Ming-Feng Yeh Chapter 6

14 Section 6-1 Example 5 (cont.) Show that T is a linear transformation from R2 to R3. 1. For any u and v in R2, we have T(u + v) = A(u + v) = Au +Av = T(u) +T(v) 2. For any u in R2 and any scalar c, we have T(cu) = A(cu) = c(Au) = cT(u) Therefore, T is a linear transformation from R2 to R3. Ming-Feng Yeh Chapter 6

15 Thm 6.2: Linear transformation given by a matrix
Section 6-1 Thm 6.2: Linear transformation given by a matrix Let A be an m  n matrix. The function T defined by T(v) = Av is a linear transformation from Rn into Rm. In order to conform to matrix multiplication with an m  n matrix, the vectors in Rn are represented by m  1 matrices and the vectors in Rm are represented by n  1 matrices. Ming-Feng Yeh Chapter 6

16 Section 6-1 Remark of Theorem 6.2 The m  n matrix zero matrix corresponds to the zero transformation from Rn into Rm. The n  n matrix identity matrix In corresponds to the identity transformation from Rn into Rn. An m  n matrix A defines a linear transformation from Rn into Rm. Ming-Feng Yeh Chapter 6

17 Ex 7: Rotation in the plane
Section 6-1 Ex 7: Rotation in the plane Show that the linear transformation T: R2  R2 given by the matrix has the property that it rotates every vector in counterclockwise about the origin through the angle . Sol: Let Ming-Feng Yeh Chapter 6

18 Ex 8: A projection in R3 The linear transformation T: R3  R3 given by
Section 6-1 Ex 8: A projection in R3 The linear transformation T: R3  R3 given by is called a projection in R3. If v = (x, y, z) is a vector in R3, then T(v) = (x, y, 0). In other words, T maps every vector in R3 to its orthogonal projection in the xy - plane. Ming-Feng Yeh Chapter 6

19 Ex 9: Linear transformation from Mm,n to Mn,m
Section 6-1 Ex 9: Linear transformation from Mm,n to Mn,m Let T: Mm,n  Mn,m be the function that maps m  n matrix A to its transpose. That is, Show that T is a linear transformation. pf: Let A and B be m  n matrix. Ming-Feng Yeh Chapter 6

20 6.2 The Kernel and Range of a Linear Transformation
Definition of Kernel of a Linear Transformation Let T:V  W be a linear transformation. Then the set of all vectors v in V that satisfy T(v) = 0 is called the kernel of T and is denoted by ker(T). The kernel of the zero transformation T: V  W consists of all of V because T(v) = 0 for every v in V. That is, ker(T) = V. The kernel of the identity transformation T: V  V consists of the single element 0. That is, ker(T) = {0}. Ming-Feng Yeh Chapter 6

21 Section 6-2 Ex 3: Finding the kernel Find the kernel of the projection T: R3  R3 given by T(x, y, z) = (x, y, 0). Sol: This linear transformation projects the vector (x, y, z) in R3 to the vector (x, y, 0) in xy-plane. Therefore, ker(T) = { (0, 0, z) : zR} Ming-Feng Yeh Chapter 6

22 Section 6-2 Ex 4: Finding the kernel Find the kernel of T: R2  R3 given by T(x1, x2) = (x1  2x2, 0, x1). Sol: The kernel of T is the set of all x = (x1, x2) in R2 s.t. T(x1, x2) = (x1  2x2, 0, x1) = (0, 0, 0). Therefore, (x1, x2) = (0, 0).  ker(T) = { (0, 0) } = { 0 } Ming-Feng Yeh Chapter 6

23 Section 6-2 Ex 5: Finding the kernel Find the kernel of T: R3  R2 defined by T(x) = Ax, where Sol: The kernel of T is the set of all x = (x1, x2, x3) in R3 s.t. T(x1, x2, x3) = (0, 0). That is, Therefore, ker(T) = {t(1, 1,1): tR} = span{(1, 1,1) } Ming-Feng Yeh Chapter 6

24 Thm 6.3: Kernel is a subspace
Section 6-2 Thm 6.3: Kernel is a subspace The kernel of a linear transformation T: V  W is a subspace of the domain V. pf: 1. ker(T) is a nonempty subset of V. 2. Let u and v be vectors in ker(T). Then T(u + v) = T(u) + T(v) = = 0 (vector addition) Thus, u + v is in the kernel 3. If c is any scalar, then T(cu) = cT(u) = c0 = 0 (scalar multiplication), Thus, cu is in the kernel. The kernel of T sometimes called the nullspace of T. Ming-Feng Yeh Chapter 6

25 Ex 6: Finding a basis for kernel
Section 6-2 Ex 6: Finding a basis for kernel Let T: R5  R4 be defined by T(x) = Ax, where x is in R5 and Find a basis for ker(T) as a subspace of R5. Ming-Feng Yeh Chapter 6

26 Example 6 (cont.) Thus one basis for the kernel T is given by
Section 6-2 Example 6 (cont.) Thus one basis for the kernel T is given by B = { (2, 1, 1, 0, 0), (1, 2, 0, 4, 1) } Ming-Feng Yeh Chapter 6

27 Section 6-2 Solution Space A basis for the kernel of a linear transformation T(x) = Ax was found by solving the homogeneous system given by Ax = 0. It is the same produce used to find the solution space of Ax = 0. Ming-Feng Yeh Chapter 6

28 Thm 6.4: Range is a subspace
Section 6-2 The Range of a Linear Transform Thm 6.4: Range is a subspace The range of a linear transformation T: V  W is a subspace of the domain W. range(T) = { T(v): v is in V } ker(T) is a subspace of V. pf: 1. range(T) is a nonempty because T(0) = 0. 2. Let T(u) and T(v) be vectors in range(T). Because u and v are in V, it follows that u + v is also in V. Hence the sum T(u) + T(v) = T(u + v) is in the range of T. (vector addition) 3. Let T(u) be a vector in the range of T and let c be a scalar. Because u is in V, it follows that cu is also in V. Hence, cT(u) = T(cu) is in the range of T. (scalar multiplication) Ming-Feng Yeh Chapter 6

29 Figure 6.6 T: V  W Kernel ker(T) is a subspace of V Codomain V W
Section 6-2 Figure 6.6 Kernel ker(T) is a subspace of V Codomain V W Domain T: V  W Range range(T) is a subspace of W Ming-Feng Yeh Chapter 6

30 Section 6-2 Column Space To find a basis for the range of a linear transformation defined by T(x) = Ax, observe that the range consists of all vectors b such that the system Ax = b is consistent. b is in the range of T if and only if b is a linear combination of the column vectors of A. Ming-Feng Yeh Chapter 6

31 Corollary of Theorems 6.3 & 6.4
Section 6-2 Corollary of Theorems 6.3 & 6.4 Let T: Rn  Rm be the linear transformation given by T(x) = Ax. [Theorem 6.3] The kernel of T is equal to the solution space of Ax = 0. [Theorem 6.4] The column space of A is equal to the range of T. Ming-Feng Yeh Chapter 6

32 Ex 7: Finding a basis for range
Section 6-2 Ex 7: Finding a basis for range Let T: R5  R4 be the linear transform given in Example 6. Find a basis for the range of T. Sol: The row echelon of A: One basis for the range of T is B = { (1, 2, 1, 0), (2, 1, 0, 0), (1, 1, 0, 2) } Ming-Feng Yeh Chapter 6

33 Rank and Nullity Let T: V  W be a linear transformation.
Section 6-2 Rank and Nullity Let T: V  W be a linear transformation. The dimension of the kernel of T is called the nullity of T and is denoted by nullity(T). The dimension of the range of T is called the rank of T and is denoted by rank(T). Ming-Feng Yeh Chapter 6

34 Thm 6.5: Sum of rank and nullity
Section 6-2 Thm 6.5: Sum of rank and nullity Let T: V  W be a linear transformation from an n-dimension vector space V into a vector space W. Then the sum of the dimensions of the range and the kernel is equal to the dimension of the domain. That is, rank(T) + nullity(T) = n or dim(range) + dim(kernel) = dim(domain) Ming-Feng Yeh Chapter 6

35 Section 6-2 Proof of Theorem 6.5 The linear transformation from an n-dimension vector space into an m-dimension vector space can be represented by a matrix, i.e., T(x) = Ax where A is an m  n matrix. Assume that the matrix A has a rank of r. Then, rank(T) = dim(range of T) = dim(column space) = rank(A) = r From Thm 4.7, we have nullity(T) = dim(kernel of T) = dim(solution space) = n  r Thus, rank(T) + nullity(T) = n + (n  r) = n Ming-Feng Yeh Chapter 6

36 Ex 8: Finding the rank & nullity
Section 6-2 Ex 8: Finding the rank & nullity Find the rank and nullity of T: R3  R3 defined by the matrix Sol: Because rank(A) = 2, the rank of T is 2. The nullity is dim(domain) – rank = 3 – 2 = 1. Ming-Feng Yeh Chapter 6

37 Ex 8: Finding the rank & nullity
Section 6-2 Ex 8: Finding the rank & nullity Let T: R5  R7 be a linear transformation Find the dimension of the kernel of T if the dimension of the range is 2. dim(kernel) = n – dim(range) = 5 – 2 = 3 Find the rank of T if the nullity of T is 4 rank(T) = n – nullity(T) = 5 – 4 = 1 Find the rank of T if ker(T) = {0} rank(T) = n – nullity(T) = 5 – 0 = 5 Ming-Feng Yeh Chapter 6

38 Section 6-2 One-to-One & Onto Linear Transformation One-to-One Mapping A linear transformation T :VW is said to be one-to-one if and only if for all u and v in V, T(u) = T(v) implies that u = v. V V W W T T One-to-one Not one-to-one Ming-Feng Yeh Chapter 6

39 Thm 6.6: One-to-one Linear transformation
Section 6-2 Thm 6.6: One-to-one Linear transformation Let T :VW be a linear transformation. Then T is one-to-one if and only if ker(T) = {0}. pf: 」Suppose T is one-to-one. Then T(v) = 0 can have only one solution: v = 0. In this case, ker(T) = {0}. 」Suppose ker(T) = {0} and T(u) = T(v). Because T is a linear transformation, it follows that T(u – v) = T(u) – T(v) = This implies that u – v lies in the kernel of T and must therefore equal 0. Hence u – v = 0 and u = v, and we can conclude that T is one-to-one. Ming-Feng Yeh Chapter 6

40 Section 6-2 Example 10 The linear transformation T: Mm,n  Mn,m given by is one-to-one because its kernel consists of only the m  n zero matrix. The zero transformation T: R3  R3 is not one-to-one because its kernel is all of R3. Ming-Feng Yeh Chapter 6

41 Onto Linear Transformation
Section 6-2 Onto Linear Transformation A linear transformation T :VW is said to be onto if every element in W has a preiamge in V. T is onto W when W is equal to the range of T. [Thm 6.7] Let T :VW be a linear transformation, where W is finite dimensional. Then T is onto if and only if the rank of T is equal to the dimension of W, i.e., rank(T) = dim(W). One-to-one: ker(T) = {0} or nullity(T) = 0 Ming-Feng Yeh Chapter 6

42 Thm 6.8: One-to-one and onto linear transformation
Section 6-2 Thm 6.8: One-to-one and onto linear transformation Let T :VW be a linear transformation with vector spaces V and W both of dimension n. Then T is one-to-one if and only if it is onto. pf: 」If T is one-to-one, then ker(T) = {0} and dim(ker(T)) = 0. In this case, dim(range of T) = n – dim(ker(T)) = n = dim(W) By Theorem 6.7, T is onto. 」If T is onto, then dim(range of T) = dim(W) = n Which by Theorem 6.5 implies that dim(ker(T)) = By Theorem 6.6, T is onto-to-one. Ming-Feng Yeh Chapter 6

43 Section 6-2 Example 11 The linear transformation T:RnRm is given by T(x) = Ax. Find the nullity and rank of T and determine whether T is one-to-one, onto, or either. T:RnRm dim(domain) rank(T) nullity(T) one-to-one onto (a) T:R3 R3 3 Yes (b) T:R2 R3 2 No (c) T:R3 R2 1 (d) T:R3 R3 Ming-Feng Yeh Chapter 6

44 Section 6-2 Isomorphisms of Vector Spaces Isomorphism Def: A linear transformation T :VW that is one-to-one and onto is called isomorphism. Moreover, if V and W are vector spaces such that there exists an isomorphism from V to W, then V and W are said to be isomorphic to each other. Theorem 6.9: Isomorphism Spaces & Dimension Two finite-dimensional vector spaces V and W are isomorphic if and only if they are of the same dimension. Ming-Feng Yeh Chapter 6

45 Ex 12: Isomorphic Vector Spaces
Section 6-2 Ex 12: Isomorphic Vector Spaces The following vector spaces are isomorphic to each other. R4 = 4-space M4,1 = space of all 4  1 matrices M2,2 = space of all 2  2 matrices P3 = space of all polynomials of degree 3 or less V = {(x1, x2, x3, x4, 0): xi is a real number} (subspace of R5) Ming-Feng Yeh Chapter 6

46 6.3 Matrices for Linear Transformation
Which one is better? The key to representing a linear transformation T:VW by a matrix is to determine how it acts on a basis of V. Once you know the image of every vector in the basis, you can use the properties of linear transformations to determine T(v) for any v in V. Simpler to write. Simpler to read, and more adapted for computer use. Ming-Feng Yeh Chapter 6

47 Thm 6.10: Standard matrix for a linear transformation
Section 6-3 Thm 6.10: Standard matrix for a linear transformation Let T: RnRm be a linear transformation such that Then the m  n matrix whose n columns corresponds to is such that T(v) = Av for every v in Rn. A is called the standard matrix for T. Ming-Feng Yeh Chapter 6

48 Section 6-3 Proof of Theorem 6.10 Let Because T is a linear transformation, we have On the other hand, Ming-Feng Yeh Chapter 6

49 Section 6-3 Example 1 Find the standard matrix for the linear transformation T: R3R2 defined by T(x, y, z) = ( x – 2y, 2x + y) Sol: Ming-Feng Yeh Chapter 6

50 Example 1 (cont.) Note that
Section 6-3 Example 1 (cont.) Note that which is equivalent to T(x, y, z) = ( x – 2y, 2x + y). Ming-Feng Yeh Chapter 6

51 Section 6-3 Example 2 The linear transformation T: R2R2 is given by projecting each point in R2 onto to the x-axis. Find the standard matrix for T. Sol: This linear transformation is given by T(x, y) = (x, 0). Therefore, the standard matrix for T is x y Ming-Feng Yeh Chapter 6

52 Composition of linear transformation
Section 6-3 Composition of Linear Transformation Composition of linear transformation The composition T, of T1: RnRm with T2: RmRp is defined by T(v) = T2( T1(v) ) = where v is a vector in Rn. The domain of T is defined to the domain of T1. The composition is not defined unless the range of T1 lies within the domain of T2. Ming-Feng Yeh Chapter 6

53 Section 6-3 Theorem 6.11 Let T1: RnRm and T2: RmRp be linear transformation with standard matrix A1 and A2. The composition T: RnRp, defined by T(v) = T2( T1(v) ), is linear transformation. Moreover, the standard matrix of A for T is given by the matrix product A = A2A1. Ming-Feng Yeh Chapter 6

54 Section 6-3 Proof of Theorem 6.11 1. Let u and v be vectors in Rn and let c be any scalar. Because T1 and T2 are linear transformation, T(u + v) = T2(T1(u + v)) = T2(T1(u) + T1(v)) = T2(T1(u)) + T2(T1(v)) = T(u) + T(v). T(cv) = T2(T1(cv)) = T2(cT1(v)) = cT2(T1(v)) = cT(v). Thus, T is a linear transformation. 2. T(v) = T2(T1(v)) = T2(A1v) = A2(A1v) = A2A1v In general, the composition is not the same as Ming-Feng Yeh Chapter 6

55 Example 3 Let T1 and T2 be linear transformation R3 from R3 such that
Section 6-3 Example 3 Let T1 and T2 be linear transformation R3 from R3 such that and Find the standard matrices for the compositions and Sol: The standard matrices for T1 and T2 are Ming-Feng Yeh Chapter 6

56 Inverse Linear Transformation
Section 6-3 Inverse Linear Transformation One benefit of matrix representation is that it can represent the inverse of a linear transformation. [Definition] If T1:RnRn and T2:RnRn are linear transformations such that T2(T1(v)) = v and T1(T2(v)) = v, then T2 is called the inverse of T1 and T1 is said to be invertible. Ming-Feng Yeh Chapter 6

57 Section 6-3 Theorem 6.12 Let T:RnRn be linear transformation with standard matrix A. Then the following conditions are equivalent. 1. T is invertible. 2. T is an isomorphism. 3. A is invertible. And, if T is invertible with standard matrix A, then the standard matrix for is Ming-Feng Yeh Chapter 6

58 Example 4 The linear transformation T: R3R3 is defined by
Section 6-3 Example 4 The linear transformation T: R3R3 is defined by Show that T is invertible, and find its inverse. Sol: A is invertible. Its inverse is Therefore T is invertible and its standard matrix is Ming-Feng Yeh Chapter 6

59 Section 6-3 Example 4 (cont.) Using the standard matrix for the inverse, we can find the rule for by computing the image of an arbitrary vector Ming-Feng Yeh Chapter 6

60 Section 6-3 Nonstandard Bases and General Vector Spaces Nonstandard Bases Finding a matrix for a linear transformation T:VW, where B and are ordered bases for V and W, respectively. The coordinate matrix of v relative to B is [v]B. To represent the linear transformation T, A must be multiplied by a coordinate matrix relative to B. The result of the multiplication will be a coordinate matrix relative to . A is called the matrix of T relative to the bases B and . Ming-Feng Yeh Chapter 6

61 Transformation Matrix
Section 6-3 Transformation Matrix Let V and W be finite-dimensional vector spaces with bases B and , respectively, where If T:VW is a linear transformation such that then the m  n matrix whose n columns correspond to is s.t. for every v in V. Ming-Feng Yeh Chapter 6

62 Example 5 Let T: R2R2 be a linear transformation defined by
Section 6-3 Example 5 Let T: R2R2 be a linear transformation defined by . Find the matrix of T relative to the bases and Sol: Therefore the coordinate matrices of T(v1) and T(v2) relative to are The matrix for T relative to B and is v1 v2 w1 w2 Ming-Feng Yeh Chapter 6

63 Example 6 For the linear transformation T: R2R2 given in Example 5,
Section 6-3 Example 6 For the linear transformation T: R2R2 given in Example 5, use the matrix A to find T(v), where v = (2, 1). Sol: Ming-Feng Yeh Chapter 6

64 6.4 Transition Matrix and Similarity
The matrix of a linear transformation T:VV depends on the basis of V. The matrix of T relative to a basis B is different from the matrix of T relative to another basis Is is possible to find a basis B such that the matrix of T relative to B is diagonal? Ming-Feng Yeh Chapter 6

65 Transition Matrices A linear transformation is defined by T:VV
Section 6-4 Transition Matrices A linear transformation is defined by T:VV Matrix of T relative to B: A Matrix of T relative to : Transition matrix from to B: P Transition matrix from B to : Ming-Feng Yeh Chapter 6

66 Example 1 Find the matrix for T: R2R2, relative to the basis
Section 6-4 Example 1 Find the matrix for T: R2R2, relative to the basis Sol: The standard matrix for T is The transition matrix from to the standard basis Therefore the matrix for T relative to is Ming-Feng Yeh Chapter 6

67 Example 2 Let and be bases for R2, and let be the matrix for
Section 6-4 Example 2 Let and be bases for R2, and let be the matrix for T: R2R2 relative to B. Find Sol: In Example 5 in Section 4.7, The matrix of T relative to is given by Ming-Feng Yeh Chapter 6

68 Section 6-4 Example 3 For the linear transformation T: R2R2 given in Example 2, find , , and for the vector v whose coordinate matrix Sol: Ming-Feng Yeh Chapter 6

69 Similar Matrices [Definition] For square matrices A and of order n,
Section 6-4 Similar Matrices Similar Matrices [Definition] For square matrices A and of order n, is said to be similar to A if there exists an invertible matrix P such that [Theorem 6.13] 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. Ming-Feng Yeh Chapter 6

70 Example 5 Suppose is the matrix for T: R3R3 relative
Section 6-4 Example 5 Suppose is the matrix for T: R3R3 relative the standard basis. Find the matrix for T relative to the basis Sol: The transition matrix from to the standard matrix is Ming-Feng Yeh Chapter 6

71 6.5 Applications of Linear Transformation
The geometry of linear transformations in the plane Reflection in the y-axis Reflection in the x-axis Ming-Feng Yeh Chapter 6

72 Reflection in the Plane
Section 6-5 Reflection in the Plane Reflection in the line y = x Ming-Feng Yeh Chapter 6

73 Expansions & Contractions in the Plane -- Horizontal
Section 6-5 Expansions & Contractions in the Plane -- Horizontal Contraction: 0 < k <1 Expansion: k >1 Ming-Feng Yeh Chapter 6

74 Expansions & Contractions in the Plane -- Vertical
Section 6-5 Expansions & Contractions in the Plane -- Vertical Contraction: 0 < k <1 Expansion: k >1 Ming-Feng Yeh Chapter 6

75 Shears in the Plane Horizontal shear: Vertical shear: Section 6-5
Ming-Feng Yeh Chapter 6


Download ppt "Chap. 6 Linear Transformations"

Similar presentations


Ads by Google