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Chap. 6 Linear Transformations

Chap. 6 Linear Transformations. Linear Algebra Ming-Feng Yeh Department of Electrical Engineering Lunghwa University of Science and Technology. 6.1 Introduction to Linear Transformation.

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Chap. 6 Linear Transformations

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  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 image of v w T: V W W: codomain of T Chapter 6

  3. Section 6-1 Map • If v is in V and w is in W s.t. T(v) = w, thenw 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 allv in V s.t. T(v) = w is called the preimage of w. Chapter 6

  4. 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 Chapter 6

  5. 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). Chapter 6

  6. 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. Chapter 6

  7. Section 6-1 Ex. 3: Not linear transformation • f(x) = sin(x)In general, sin(x1 + x2)  sin(x1) + sin(x2) • f(x) = x2In general, • f(x) = x + 1f(x1 + x2) = x1 + x2 + 1f(x1) + f(x2) = (x1 + 1) + (x2 + 1) = x1 + x2 + 2Thus, f(x1 + x2)  f(x1) + f(x2) Chapter 6

  8. 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 Chapter 6

  9. 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, thenT(v) = c1T(v1) + c2T(v2) + … + cnT(vn) Chapter 6

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

  12. 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) Chapter 6

  13. 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) Chapter 6

  14. Section 6-1 Example 5 (cont.) • Show that T is a linear transformation from R2to R3. 1. For any u and v in R2, we haveT(u + v) = A(u + v) = Au +Av = T(u) +T(v) 2. For any u in R2 and any scalar c, we haveT(cu) = A(cu) = c(Au) = cT(u) Therefore, T is a linear transformation from R2to R3. Chapter 6

  15. Section 6-1 Thm 6.2: Linear transformation given by a matrix Let A be an m n matrix. The function T defined byT(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. 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 matrixIn corresponds to the identity transformation from Rn into Rn. • An m n matrix A defines a linear transformation from Rn into Rm. Chapter 6

  17. Section 6-1 Ex 7: Rotation in the plane Show that the linear transformation T: R2 R2given by the matrix has the property that it rotates every vector in counterclockwiseabout the origin through the angle . Sol:Let Chapter 6

  18. Section 6-1 Ex 8: A projection in R3 The linear transformation T: R3 R3given 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. Chapter 6

  19. 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. Chapter 6

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

  21. Section 6-2 Ex 3: Finding the kernel Find the kernel of the projection T: R3 R3given by T(x, y, z) = (x, y, 0). Sol: This linear transformationprojects the vector (x, y, z)in R3 to the vector (x, y, 0)in xy-plane. Therefore,ker(T) = { (0, 0, z) : zR} Chapter 6

  22. Section 6-2 Ex 4: Finding the kernel Find the kernel of T: R2 R3given 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 } Chapter 6

  23. Section 6-2 Ex 5: Finding the kernel Find the kernel of T: R3 R2defined 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) } Chapter 6

  24. 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). ThenT(u + v) = T(u) + T(v) = 0 + 0 = 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. Chapter 6

  25. 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. Chapter 6

  26. 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) } 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. Chapter 6

  28. 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 sumT(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) Chapter 6

  29. Section 6-2 Figure 6.6 Kernel ker(T) is a subspace of V Codomain V 0 W Domain T: V W Range range(T) is a subspace of W 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. Chapter 6

  31. 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 kernelof 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. Chapter 6

  32. 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) } Chapter 6

  33. 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 therank of T and is denoted by rank(T). Chapter 6

  34. 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) = nordim(range) + dim(kernel) = dim(domain) 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 nmatrix. 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 Chapter 6

  36. 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. Chapter 6

  37. 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 4rank(T) = n – nullity(T) = 5 – 4 = 1 • Find the rank of T if ker(T) = {0}rank(T) = n – nullity(T) = 5 – 0 = 5 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 beone-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 Not one-to-one One-to-one Chapter 6

  39. 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 thatT(u – v) = T(u) – T(v) = 0 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. 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 nzero matrix. • The zero transformationT: R3 R3 is notone-to-one because its kernel is all of R3. Chapter 6

  41. 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 Chapter 6

  42. 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 Wboth 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)) = 0 By Theorem 6.6, T is onto-to-one. 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. 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 & DimensionTwo finite-dimensional vector spaces V and W are isomorphic if and only if they are of the same dimension. Chapter 6

  45. 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) Chapter 6

  46. 6.3 Matrices for Linear Transformation • Which one is better? • The key to representing a linear transformation T:VWby 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 anyv in V. Simpler to write. Simpler to read, and more adapted for computer use. Chapter 6

  47. 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 ncolumns corresponds to is such that T(v) = Av for every v in Rn. A is called the standard matrix for T. Chapter 6

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

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

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