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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 Learn about functions that map a vector space V into a vector space W --- T: V W V: domain of T range v T: V W w image of v W: codomain of T Ming-Feng Yeh Chapter 6 6-2 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 6-3 Section 6-1 Ex. 1: A function from 2 R into 2 R 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 6-4 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 6-5 Section 6-1 Ex. 2: Verifying a linear 2 2 transformation from R into R 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 6-6 Section 6-1 Ex. 3: Not linear transformation f(x) = sin(x) In general, sin(x1 + x2) sin(x1) + sin(x2) f(x) = x2 2 2 2 In general, ( x1 x2 ) x1 x2 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 6-7 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 6-8 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 6-9 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 6-10 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 6-11 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 6-12 Section 6-1 Ex 5: Linear transformation defined by a matrix The function T: R2 R3 is defined as follows 0 3 v1 T ( v) Av 2 1 v2 1 2 Find T(v), where v = (2, 1) 0 3 6 2 T ( v) Av 2 1 3 1 1 2 0 Therefore, T(2, 1) = (6, 3, 0) Ming-Feng Yeh Chapter 6 6-13 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 6-14 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 6-15 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. a11 a12 a1n v1 a11v1 a12v2 a1nvn a v a v a v a v a a 22 2n 2 22 2 2n n Av 21 21 1 am1 am 2 amn vn am1v1 am 2v2 amnvn Rm Rn Ming-Feng Yeh Chapter 6 6-16 Section 6-1 Ex 7: Rotation in the plane Show that the linear transformation T: R2 R2 given by the cos sin matrix A has the property that it rotates sin cos every vector in counterclockwise about the origin through the angle . Sol: Let v ( x, y ) (r cos , r sin ) cos sin r cos y T ( v ) Av r sin sin cos T ( x, y ) r cos cos r sin sin r sin cos r cos sin r cos( ) r sin( ) Ming-Feng Yeh Chapter 6 ( x, y ) x 6-17 Section 6-1 Ex 8: A projection in 3 R The linear transformation T: R3 R3 given by z 1 0 0 A 0 1 0 ( x, y , z ) 0 0 0 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, x T ( x, y , z ) T maps every vector in R3 to its ( x , y ,0 ) orthogonal projection in the xy - plane. Ming-Feng Yeh Chapter 6 6-18 y 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, T ( A) AT Show that T is a linear transformation. pf: Let A and B be m n matrix. T ( A B) ( A B)T AT BT T ( A) T ( B ) and T (cA) (cA)T c( AT ) cT ( A) T is a linear tra nsformatio n form M m , n into M n , m Ming-Feng Yeh Chapter 6 6-19 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 6-20 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). z Sol: This linear transformation projects the vector (x, y, z) (0,0, z ) in R3 to the vector (x, y, 0) in xy-plane. Therefore, (0,0,0) ker(T) = { (0, 0, z) : zR} x Ming-Feng Yeh Chapter 6 y 6-21 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 6-22 Section 6-2 Ex 5: Finding the kernel Find the kernel of T: R3 R2 defined by T(x) = Ax, where 1 1 2 A 1 2 3 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, x1 x1 t 1 1 2 0 x2 x2 t , t R 1 2 3 0 x3 x3 t Therefore, ker(T) = {t(1, 1,1): tR} = span{(1, 1,1) } Ming-Feng Yeh Chapter 6 6-23 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 + 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. Ming-Feng Yeh Chapter 6 6-24 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 1 2 and A 1 0 2 0 1 3 0 2 0 0 1 1 1 0 . 0 1 2 8 Find a basis for ker(T) as a subspace of R5. Ming-Feng Yeh Chapter 6 6-25 Section 6-2 Example 6 (cont.) 1 2 1 0 2 0 1 1 3 1 0 2 0 0 0 2 x1 0 x1 2 1 1 1 2 x x 0 2 2 0 x3 0 x3 s 1 t 0 1 0 4 x4 0 x4 8 x5 0 x5 0 1 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 6-26 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 6-27 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 6-28 Section 6-2 Figure 6.6 Kernel ker(T) is a subspace of V Codomain V Domain T: V W Range Ming-Feng Yeh 0 W range(T) is a subspace of W Chapter 6 6-29 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. a11 a12 a a22 21 Ax am1 am 2 a1n x1 a11 a1n b1 a a b a2 n x2 x1 21 xn 2 n 2 b amn xn am1 amn bm 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 6-30 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 6-31 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: 0 1 1 1 0 2 1 2 2 1 0 1 1 3 1 0 A 1 0 2 0 1 0 0 0 0 2 8 0 0 0 0 0 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 0 1 0 2 1 4 0 0 6-32 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 6-33 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 6-34 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 6-35 Section 6-2 Ex 8: Finding the rank & nullity Find the rank and nullity of T: R3 R3 defined by 1 0 2 0 1 1 A the matrix 0 0 0 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 6-36 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 6-37 Section 6-2 One-to-One & Onto Linear Transformation One-to-One Mapping A linear transformation T :VW 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 Not one-to-one One-to-one Ming-Feng Yeh Chapter 6 6-38 Section 6-2 Thm 6.6: One-to-one Linear transformation Let T :VW 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) = 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. Ming-Feng Yeh Chapter 6 6-39 Section 6-2 Example 10 The linear transformation T: Mm,n Mn,m given by T ( A) AT is one-to-one because its kernel consists of only the m n zero matrix. The zero transformation T: R3 R3 is not oneto-one because its kernel is all of R3. Ming-Feng Yeh Chapter 6 6-40 Section 6-2 Onto Linear Transformation A linear transformation T :VW 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 :VW 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 6-41 Section 6-2 Thm 6.8: One-to-one and onto linear transformation Let T :VW 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)) = 0 By Theorem 6.6, T is onto-to-one. Ming-Feng Yeh Chapter 6 6-42 Section 6-2 Example 11 The linear transformation T:RnRm is given by T(x) = Ax. Find the nullity and rank of T and determine whether T is one-to-one, onto, or either. 1 2 0 1 2 1 2 0 1 2 0 0 1 1 (a) A 0 1 1, (b) A 0 1, (c) A , ( d ) A 0 1 1 0 0 1 0 0 0 0 0 T:RnRm dim(domain) rank(T) nullity(T) one-to-one onto (a) T:R3 R3 3 3 0 Yes Yes (b) T:R2 R3 2 2 0 Yes No (c) T:R3 R2 3 2 1 No Yes (d) T:R3 R3 3 2 1 No No Ming-Feng Yeh Chapter 6 6-43 Section 6-2 Isomorphisms of Vector Spaces Isomorphism Def: A linear transformation T :VW that is one-toone 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 6-44 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 6-45 6.3 Matrices for Linear Transformation Which one is better? T ( x1 , x2 , x3 ) (2 x1 x2 x3 , x1 3x2 2 x3 , 3x2 4 x3 ) 2 1 1 x1 Simpler to write. Simpler T (x) Ax 1 3 2 x2 to read, and more adapted 0 3 4 x3 for computer use. The key to representing a linear transformation T:VW 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. Ming-Feng Yeh Chapter 6 6-46 Section 6-3 Thm 6.10: Standard matrix for a linear transformation Let T: RnRm be a linear transformation such that 1 a11 0 a12 0 a1n 0 a 1 a 0 a T (e1 ) T ( ) 21 , T (e2 ) T ( ) 22 , , T (en ) T ( ) 2 n a a 0 m1 0 m 2 1 amn Then the m n matrix whose n columns corresponds to T (ei ), a11 a12 an1 is such that T(v) = Av for every a a a v in Rn. A is called the standard 22 n2 A 21 matrix for T. am1 am 2 amn Ming-Feng Yeh Chapter 6 6-47 Section 6-3 Proof of Theorem 6.10 Let v v1 v2 vn v1e1 v2e2 vnen Rn . Because T is a linear transformation, we have T ( v ) T (v1e1 v2e 2 vne n ) T (v1e1 ) T (v2e 2 ) T (vne n ) v1T (e1 ) v2T (e 2 ) vnT (e n ) On the other hand, a11 a12 an1 v1 a11 a12 a1n a v a a a a a 22 n2 2 Av 21 v1 21 v2 22 vn 2 n a a a v a a m2 mn n m1 m1 m2 amn v1T (e1 ) v2T (e 2 ) vnT (e n ) T ( v) T Ming-Feng Yeh Chapter 6 6-48 Section 6-3 Example 1 Find the standard matrix for the linear transformation T: R3R2 defined by T(x, y, z) = ( x – 2y, 2x + y) 1 Sol: 1 T (e1 ) T (1, 0, 0) (1, 2) T (e1 ) T 0 0 2 0 2 T (e 2 ) T (0, 1, 0) (2, 1) T (e 2 ) T 1 0 1 0 0 T (e3 ) T (0, 0, 1) (0, 0) T (e3 ) T 0 1 0 Ming-Feng Yeh Chapter 6 6-49 Section 6-3 Example 1 (cont.) 1 2 0 A T (e1 )T (e2 )T (e3 ) 2 1 0 Note that x x 1 2 0 x 2 y A y y 2 1 0 2 x y z z which is equivalent to T(x, y, z) = ( x – 2y, 2x + y). Ming-Feng Yeh Chapter 6 6-50 Section 6-3 Example 2 The linear transformation T: R2R2 is given by projecting each point in R2 onto to the x-axis. Find the standard y matrix for T. Sol: This linear transformation is given by ( x, y ) T(x, y) = (x, 0). Therefore, the standard matrix for T is 1 0 A T (1, 0)T (0, 1) 0 0 (x, 0) Ming-Feng Yeh Chapter 6 6-51 x Section 6-3 Composition of Linear Transformation Composition of linear transformation The composition T, of T1: RnRm with T2: RmRp is defined by T(v) = T2( T1(v) ) = T2 T1 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. T2 T1 R n v Rm w Rp u T Ming-Feng Yeh Chapter 6 6-52 Section 6-3 Theorem 6.11 Let T1: RnRm and T2: RmRp be linear transformation with standard matrix A1 and A2. The composition T: RnRp, 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 6-53 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 T2 T1 is not the same as T1 T2 . Ming-Feng Yeh Chapter 6 6-54 Section 6-3 Example 3 Let T1 and T2 be linear transformation R3 from R3 such that T1 ( x, y, z) (2x y, 0, x z) and T2 ( x, y, z) ( x y, z, y) Find the standard matrices for the compositions T T2 T1 and T T1 T2 . Sol: The standard matrices for T1 and T2 are 2 1 0 1 1 0 A1 0 0 0, A2 0 0 1 1 0 1 0 1 0 2 1 0 2 2 1 T A A1 A2 1 0 1, T A A2 A1 0 0 0 0 0 0 1 0 0 Ming-Feng Yeh Chapter 6 6-55 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:RnRn and T2:RnRn 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 6-56 Section 6-3 Theorem 6.12 Let T:RnRn 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 1 1 T A the standard matrix for is . Ming-Feng Yeh Chapter 6 6-57 Section 6-3 Example 4 The linear transformation T: R3R3 is defined by T ( x, y , z ) ( 2 x 3 y z , 3 x 3 y z , 2 x 4 y z ) Show that T is invertible, and find its inverse. 2 3 1 Sol: A 3 3 1 det( A) 0 1 0 2 4 1 1 A is invertible. Its inverse is A1 1 0 1 6 2 3 1 Therefore T is invertible and its standard matrix is A . Ming-Feng Yeh Chapter 6 6-58 Section 6-3 Example 4 (cont.) Using the standard matrix for the inverse, we can find the rule for T 1 by computing the image of an arbitrary vector v ( x, y, z ). 1 0 x x y 1 T 1 ( v) 1 0 1 y x z 6 2 3 z 6 x 2 y 3 y T 1 ( x, y, z ) ( x y, x z, 6 x 2 y 3z ). Ming-Feng Yeh Chapter 6 6-59 Section 6-3 Nonstandard Bases and General Vector Spaces Nonstandard Bases Finding a matrix for a linear transformation T:VW, where B and B 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 B . [T ( v)]B A[ v]B . A is called the matrix of T relative to the bases B and B . Ming-Feng Yeh Chapter 6 6-60 Section 6-3 Transformation Matrix Let V and W be finite-dimensional vector spaces with bases B and B , respectively, where B {v1 , v 2 , ..., v n } If T:VW is a linear transformation such that a11 a12 a1n a a a [T ( v1 )]B 21 , [T ( v 2 )]B 22 , ,[T ( v n )]B 2 n , a a mn2 columns amn to then the mmn1 matrix whose correspond a11 a12 a a22 21 A am1 am 2 Ming-Feng Yeh an1 an 2 is s.t. , [T (every v)]B vAin [ v]V. for B amn Chapter 6 [T ( v)]B , 6-61 Section 6-3 Example 5 Let T: R2R2 be a linear transformation defined by T ( x, y ) ( x y, 2 x y ). Find the matrix of T relative to the bases B {(1, 2), (1, 1)} and B {(1, 0), (0, 1)} v1 v2 w1 w2 Sol: T ( v1 ) T (1, 2) (3, 0) 3w1 0w 2 T ( v 2 ) T (1, 1) (0, 3) 0w1 3w 2 Therefore the coordinate matrices of T(v1) and T(v2) relative to B are [T ( v1 )]B 3 0T , [T ( v 2 )]B 0 3T . 3 0 The matrix for T relative to B and B is A 0 3 Ming-Feng Yeh Chapter 6 6-62 Section 6-3 Example 6 For the linear transformation T: R2R2 given in Example 5, use the matrix A to find T(v), where v = (2, 1). Sol: B {(1, 2), (1, 1)} 1 v (2, 1) 1(1, 2) 1(1, 1) [ v]B 1 0 1 3 3 [T ( v)]B A[ v]B 3 3 1 3 B {(1, 0), (0, 1)} T ( v) 3(1, 0) 3(0, 1) (3, 3). Ex 5 : T ( x, y ) ( x y, 2 x y ) T (2, 1) (3, 3) Ming-Feng Yeh Chapter 6 6-63 6.4 Transition Matrix and Similarity The matrix of a linear transformation T:VV 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 B Is is possible to find a basis B such that the matrix of T relative to B is diagonal? Ming-Feng Yeh Chapter 6 6-64 Section 6-4 Transition Matrices A linear transformation is defined by T:VV Matrix of T relative to B: A T ( v)B Av B Matrix of T relative to B : A T ( v)B Av B Transition matrix from B to B: P 1 1 [ v ]B P[ v ]B ; [ v ]B P [ v ]B Transition matrix from B to B : P [v]B A P 1 P [v]B Ming-Feng Yeh [T ( v)]B A [T ( v)]B Chapter 6 T ( v)B AvB 1 T ( v)B P APvB A P 1 AP 6-65 Section 6-4 Example 1 Find the matrix A for T: R2R2, T ( x, y ) (2 x 2 y, x 3 y ) relative to the basis B {(1, 0), (1, 1)}. 2 2 Sol: The standard matrix for T is A 1 3 The transition matrix from B to the standard basis 1 1 1 1 1 B {(1, 0), (0, 1)} is P P 0 1 0 1 Therefore the matrix for T relative to B is 3 2 A P AP 1 2 1 Ming-Feng Yeh Chapter 6 6-66 Section 6-4 Example 2 Let B {( 3, 2), (4, 2)} and B {( 1, 2), (2, 2)}. be bases for R2, 2 7 and let A be the matrix for 3 7 T: R2R2 relative to B. Find A. Sol: In Example 5 in Section 4.7, 3 2 1 1 2 P ,P 2 1 2 3 The matrix of T relative to B is given by 2 1 1 A P AP 1 3 Ming-Feng Yeh Chapter 6 6-67 Section 6-4 Example 3 For the linear transformation T: R2R2 given in Example 2, find [v]B , [T ( v)]B , and [T ( v)]B for the vector v whose T coordinate matrix [ v]B 3 1 Sol: 3 [ v]B P[ v]B 2 2 [T ( v)]B A[ v]B 3 2 3 7 1 1 5 7 7 21 7 5 14 1 2 21 7 [T ( v)]B P [T ( v)]B 2 3 14 0 A[ v]B 1 Ming-Feng Yeh Chapter 6 6-68 Section 6-4 Similar Matrices Similar Matrices [Definition] For square matrices A and A of order n, A is said to be similar to A if there exists an invertible matrix 1 P such that A P AP. [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 6-69 Section 6-4 Example 5 1 3 0 Suppose A 3 1 0 is the matrix for T: R3R3 relative 0 0 2 the standard basis. Find the matrix for T relative to the basis B {(1, 1, 0), (1, 1, 0), (0, 0, 1)} Sol: The transition matrix from B to the standard matrix is 12 12 1 1 0 P 1 1 0 P 1 12 12 0 0 0 0 1 Ming-Feng Yeh 0 0 4 0 0 A P 1 AP 0 2 0 0 0 2 1 Chapter 6 6-70 6.5 Applications of Linear Transformation The geometry of linear transformations in the plane Reflection in the y-axis T ( x, y ) ( x , y ) Reflection in the x-axis T ( x, y ) ( x, y ) 1 0 x x 0 1 y y y ( x, y ) ( x, y ) 1 0 x x 0 1 y y y ( x, y ) x x Ming-Feng Yeh Chapter 6 ( x, y ) 6-71 Section 6-5 Reflection in the Plane Reflection in the line y = x y T ( x, y ) ( y , x ) 0 1 x y 1 0 y x ( y, x) ( x, y ) x yx Ming-Feng Yeh Chapter 6 6-72 Section 6-5 Expansions & Contractions in the Plane -- Horizontal T ( x, y ) (kx, y ) y ( kx, y ) k 0 x kx 0 1 y y y ( x, y ) ( x, y ) x x Contraction: 0 < k <1 Ming-Feng Yeh ( kx, y ) Expansion: k >1 Chapter 6 6-73 Section 6-5 Expansions & Contractions in the Plane -- Vertical T ( x, y ) ( x, ky) y 1 0 x x 0 k y ky y ( x, y ) ( kx, y ) ( kx, y ) ( x, y ) x Contraction: 0 < k <1 Ming-Feng Yeh x Expansion: k >1 Chapter 6 6-74 Section 6-5 Shears in the Plane Horizontal shear: T ( x, y ) ( x ky, y ) Vertical shear: T ( x, y ) ( x, y kx) 1 k x x ky 0 1 y y y ( x, y ) ( x ky, y ) 1 0 x x k 1 y kx y y ( x, kx y ) ( x, y ) x x Ming-Feng Yeh Chapter 6 6-75