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Elementary Linear Algebra Anton & Rorres, 9 th Edition

Elementary Linear Algebra Anton & Rorres, 9 th Edition. Lecture Set – 08 Chapter 8: Linear Transformations. Chapter Content. General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism.

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Elementary Linear Algebra Anton & Rorres, 9 th Edition

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  1. Elementary Linear AlgebraAnton & Rorres, 9th Edition Lecture Set – 08 Chapter 8: Linear Transformations

  2. Chapter Content • General Linear Transformations • Kernel and Range • Inverse Linear Transformations • Matrices of General Linear Transformations • Similarity • Isomorphism Elementary Linear Algebra

  3. Linear Transformation • Definition • If T : V  W is a function from a vector space V into a vector space W, then T is called a linear transformationfrom V to W if for all vectors u and v in V and all scalars c • T (u + v) = T (u) + T (v) • T (cu) = cT (u) In the special case where V = W, the linear transformation T : V  V is called a linear operatoron V. Elementary Linear Algebra

  4. Linear Transformation • Example (Zero Transformation) • The mapping T : V  W such that T(v) = 0 for every v in V is a linear transformation called the zero transformation. • Example (Identity Operator) • The mapping I : V  I defined by I (v) = v is called the identity operatoron V. Elementary Linear Algebra

  5. Orthogonal Projections • Suppose that W is a finite-dimensional subspace of an inner product spaceV ; then the orthogonal projectionof V onto W is the transformation defined by T (v) = projWv • If S = {w1, w2, …, wr} is any orthogonal basis for W, then T (v) is given by the formula T (v) = projWv = v,w1 w1 + v,w2 w2 + ··· + v,wr wr • This projection a linear transformation: • T(u + v) = T(u) + T(v) • T(cu) = cT(u) Elementary Linear Algebra

  6. A Linear Transformation from a Space V to Rn • Let S = {w1, w2, …, wn} be a basis for an n-dimensional vector space V, and let (v)s = (k1,, k2,, …,kn) be the coordinate vector relative to S of a vector v in V; thus v = k1w1+k2w2 +…+ kn wn • DefineT : V  Rn to be the function that maps v into its coordinate vector relative to S; that is, T (v) = (v)s = (k1,, k2,, …,kn) • Then the function T is a linear transformation: • Let u = c1w1+c2w2 +…+ cn wnand v = d1w1+d2w2 +…+ dn wn • Check if (u + v)s = (u)s + (v)s and (ku)s = k(u)s Elementary Linear Algebra

  7. A Linear Transformation fromPnto Pn+1 • Let p = p(x) = c0+ c1x+ ··· + cnx n be a polynomial in Pn , and define the function T : Pn  Pn+1by T (p) = T (p(x)) = xp(x) = c0x + c1x2+ ··· + cnx n+1 • The function T is a linear transformation: • For any scalar k and any polynomials p1 and p2 in Pn we have • T (p1 + p2) =T (p1(x) + p2 (x))= x (p1(x) + p2 (x))= xp1(x) + x p2 (x) = T (p1) + T (p2) • T (k p) = T (k p(x))= x (k p(x)) = k (x p(x))= k T(p) Elementary Linear Algebra

  8. A Linear Transformation Using an Inner Product • Let V be an inner product space and let v0 be any fixed vector in V. Let T : V  R be the transformation that maps a vector v into its inner product with v0; that is, T (v) = v, v0 • From the properties of an inner product • T (u + v) = u + v, v0 = u, v0 + v, v0 • T (k u) = k u, v0 = k u, v0 = kT (u) • Thus, T is a linear transformation. Elementary Linear Algebra

  9. Properties of Linear Transformation • If T : V  W is a linear transformation, then for any vectors v1 and v2 in V and any scalars c1and c2, we have T (c1v1 +c2v2) = T (c1v1) + T (c2v2) = c1T (v1) + c2T (v2) • More generally, if v1 ,v2 ,…, vn are vectors in V and c1,c2 ,…, cn are scalars, then T (c1v1 +c2v2 +…+ cnvn ) = c1T (v1)+c2T (v2)+…+ cnT(vn) • The above equation is sometimes described by saying that linear transformations preserve linear combinations. Elementary Linear Algebra

  10. Theorem • Theorem 8.1 • If T : V W is a linear transformation, then • T(0) = 0 • T(-v) = -T(v) for all v in V • T(v – w) = T(v) – T(w) for all v and w in V Elementary Linear Algebra

  11. Finding Linear Transformations from Images of Basis • If T : V  W is a linear transformation, and if {v1 ,v2 ,…, vn} is any basis for V, then the image T (v) of any vector v in V can be calculated from the images T (v1), T (v2), …, T (vn) of the basis vectors. • This can be done by first expressing v as a linear combination of the basis vectors, say v = c1 v1+ c2 v2+…+ cn vn and then the transformation becomes T (v) = c1 T (v1) + c2 T (v2) + … + cn T (vn) • A linear transformation is completely determined by its images of any basis vectors. Elementary Linear Algebra

  12. Example • Consider the basis S = {v1 ,v2 , v3} for R3 , where v1 = (1,1,1), v2 = (1,1,0), and v3 = (1,0,0). Let T: R3 R2be the linear transformation such that T (v1) = (1,0), T (v2) = (2,-1), T (v3) = (4,3). Find a formula for T (x1,x2, x3); then use this formula to compute T (2, -3, 5). Elementary Linear Algebra

  13. Composition of T2 with T1 • Definition • If T1: U  V and T2 : V  W are linear transformations, the composition of T2 with T1, denoted by T2 T1 (read “T2 circle T1 ”), is the function defined by the formula (T2 T1 )(u) = T2 (T1 (u)) where u is a vector in U. • Theorem 8.1.2 • If T1: U  V and T2 : V  W are linear transformations, then (T2 T1 ) : U  W is also a linear transformation. Elementary Linear Algebra

  14. Remark • The compositions can be defined for more than two linear transformations. • For example, if T1 : U V and T2:V  W,and T3:W  Y are linear transformations, then the composition T3  T2 T1 is defined by (T3 T2 T1 )(u) = T3(T2(T1 (u))) Elementary Linear Algebra

  15. Chapter Content • General Linear Transformations • Kernel and Range • Inverse Linear Transformations • Matrices of General Linear Transformations • Similarity • Isomorphism Elementary Linear Algebra

  16. Kernel and Range • Recall: • If A is an mn matrix, then the nullspace of A consists of all vector x in Rn such that Ax = 0. • The column space of A consists of all vectors b in Rm for which there is at least one vector x in Rn such that Ax = b. • The nullspace of A consists of all vectors in Rn that multiplication by A maps into 0. (in terms of matrix transformation) • The column space of A consists of all vectors in Rm that are images of at least one vector in Rn under multiplication by A. (in terms of matrix transformation) Elementary Linear Algebra

  17. Kernel and Range • Definition • If T : V  W is a linear transformation, then the set of vectors in V that T maps into 0 is called the kernel of T; it is denoted by ker(T). • The set of all vectors in W that are images under T of at least one vector in V is called the range of T; it is denoted by R(T). Elementary Linear Algebra

  18. Examples • If TA : Rn Rmis multiplication by the mn matrix A, then the kernel of TA is the nullspace of A and the range of TA is the column space of A. • Let T : V  W be the zero transformation. Since T maps every vector in V into 0, it follows that ker(T) = V. Moreover, since 0 is the only image under T of vectors in V, we have R(T) = {0}. • Let I : V  V be the identity operator. Since I (v) = v for all vectors in V, every vector in V is the image of some vector; thus, R(I) = V. Since the only vector that I maps into 0 is 0, it follows ker(I) = {0}. Elementary Linear Algebra

  19. Example • Let T : R3R3 be the orthogonal projection on the xy-plane. The kernel of T is the set of points that T maps into 0 = (0,0,0); these are the points on the z-axis. • Since T maps every points in R3 into the xy-plane, the range of T must be some subset of this plane. But every point (x0 ,y0 ,0) in the xy-plane is the image under T of some point. Thus R(T) is the entire xy-plane. Elementary Linear Algebra

  20. Example • Let T : R2R2 be the linear operator that rotates each vector in the xy-plane through the angle . • Since every vector in the xy-plane can be obtained by rotating through some vector through angle , we have R(T) = R2. • The only vector that rotates into 0 is 0, so ker(T) = {0}. Elementary Linear Algebra

  21. Properties of Kernel and Range • Theorem 8.2.1 • If T : V  W is linear transformation, then: • The kernel of T is a subspace of V. • The range of T is a subspace of W. Elementary Linear Algebra

  22. Properties of Kernel and Range • Definition • If T : V  W is a linear transformation, then the dimension of the range of T is called the rank of T and is denoted by rank(T). • The dimension of the kernel is called the nullity of T and is denoted by nullity(T). • Theorem 8.2.2 • If A is an mn matrix and TA : Rn Rm is multiplication by A, then: • nullity (TA) = nullity (A) • rank (TA) = rank (A) Elementary Linear Algebra

  23. Example • Let TA : R6R4 be multiplication byFind the rank and nullity of TA • In Example 1 of Section 5.6 we showed that rank (A) = 2 and nullity (A) = 4. (use reduced row-echelon form, etc.) • Thus, from Theorem 8.2.2, rank (TA) = 2 and nullity (TA) = 4. Elementary Linear Algebra

  24. Example • Let T : R3R3 be the orthogonal projection on the xy-plane. • From Example 4, the kernel of T is the z-axis, which is one-dimensional; and the range of T is the xy-plane, which is two-dimensional. • Thus, nullity (T) = 1 and rank (T) = 2. Elementary Linear Algebra

  25. Dimension Theorem for Linear Transformations • Theorem 8.2.3 • If T : V W is a linear transformation from an n-dimensional vector space V to a vector space W, then rank(T) + nullity(T) = n • Remark • In words, this theorem states that for linear transformations the rank plus the nullity is equal to the dimension of the domain. Elementary Linear Algebra

  26. Chapter Content • General Linear Transformations • Kernel and Range • Inverse Linear Transformations • Matrices of General Linear Transformations • Similarity • Isomorphism Elementary Linear Algebra

  27. One-to-One Linear Transformation • A linear transformation T : V W is said to be one-to-one if T maps distinct vectors in V into distinct vectors in W. • Examples • If A is an nn matrix and TA: Rn  Rnis multiplication by A, then TAis one-to-one if and only if A is an invertible matrix (Theorem 4.3.1). Elementary Linear Algebra

  28. Example • Let T : R2 R2 be the linear operator that rotates each vector in the xy-plane through an angle . We showed that ker(T) = {0} and R(T) = R2. • Thus, rank(T) + nullity(T) = 2 + 0 = 2. Elementary Linear Algebra

  29. Theorem 8.3.1 (Equivalent Statements) • If T : V W is a linear transformation, then the following are equivalent. • T is one-to-one • The kernel of T contains only zero vector; that is, ker(T) = {0} • Nullity(T) = 0 Elementary Linear Algebra

  30. Theorem 8.3.2 • If V is a finite-dimensional vector space and T : V V is a linear operator, then the following are equivalent. • T is one-to-one • ker(T) = {0} • Nullity(T) = 0 • The range of T is V; that is, R(T) = V Elementary Linear Algebra

  31. Example • Let TA : R4R4 be multiplication byDetermine whether TA is one to one. • Solution: • det(A) = 0, since the first two rows of A are proportional A is not invertibleTA is not one-to-one. Elementary Linear Algebra

  32. Inverse Linear Transformations • If T : VW is a linear transformation, then the range of T denoted by R (T), is the subspace of W consisting of all images under T of vectors in V. • If T is one-to-one, then each vector w in R(T) is the image of a unique vector v in V. • This uniqueness allows us to define a new function, call the inverse of T, denoted by T –1, which maps w back into v. • The mapping T –1 : R (T) V is a linear transformation. Moreover, T–1(T (v)) = T–1(w) = v T–1(T (w)) = T–1(v) = w Elementary Linear Algebra

  33. Inverse Linear Transformations • If T : VW is a one-to-one linear transformation, then the domain of T –1 is the range of T. • The range may or may not be all of W (one-to-one but notonto). • For the special case that T : VV, then the linear transformation is one-to-one and onto. Elementary Linear Algebra

  34. Example (An Inverse Transformation) • Let T : R3 R3 be the linear operator defined by the formula T (x1, x2, x3) = (3x1 + x2, -2x1 – 4x2 + 3x3, 5x1 + 4x2 – 2x3). Determine whether T is one-to-one; if so, find T -1(x1,x2,x3) . • Solution: Elementary Linear Algebra

  35. Theorem 8.3.3 • If T1 : U V and T2 : V W are one to one linear transformation then: • T2 T1 is one to one • (T2 T1)-1 = T1-1 T2-1 Elementary Linear Algebra

  36. Chapter Content • General Linear Transformations • Kernel and Range • Inverse Linear Transformations • Matrices of General Linear Transformations • Similarity • Isomorphism Elementary Linear Algebra

  37. Matrices of General Linear Transformations • Remark: • If V and W are finite-dimensional vector spaces (not necessarily Rn and Rm), then any transformation T : VW can be regarded as a matrix transformation. • The basic idea is to work with coordinate matrices of the vectors rather than with the vectors themselves. Elementary Linear Algebra

  38. A vector in V (n-dimensional) A vector in W (m-dimensional) T T (x) x A vector in Rn A vector in Rm ? [T (x)]B [x]B Matrices of Linear Transformations • Suppose V and W are n and m dimensional vector space and B and B are bases for V and W, then for x in V, the coordinate matrix [x]B will be a vector in Rn, and coordinate matrix [T(x)] B will be a vector in Rm . Elementary Linear Algebra

  39. If we let A be the standard matrix for this transformation, then A [x]B = [T (x)]B The matrix A is called the matrix for T with respect to the bases B and B T maps V into W T T (x) x [T (x)]B [x]B A Multiplication by Amaps Rn to Rm Matrices of Linear Transformations Elementary Linear Algebra

  40. Matrices of Linear Transformations • Let B = {u1, …, un} be a basis for the n-dimensional space V and B = {u1, …, um} be a basis for the m-dimensional space W. • Consider an mn matrix such that A [x]B = [T(x)]B holds for all vectors x in V. • That is, A [x]B = [T(x)]B has to hold for the basis vectorsu1, …, un. • Thus, we need A [u1]B = [T(u1)]B , A [u2]B = [T(u2)]B , …, A [un]B = [T(un)]B • Since [u1]B = e1 , [u2]B = e2 , …, [un]B = en Elementary Linear Algebra

  41. Matrices of Linear Transformations • We have • Thus, , which is the matrix for T w.r.t. the bases B and B, and denoted by the symbol [T]B,B • That is, and Basis for the image space Basis for the domain Elementary Linear Algebra

  42. Matrices for Linear Operators • In the special case where V = W, the resulting matrix is called the matrix for T with respect to the basis B and denoted by [T]B rather than [T]B,B. • If B = {u1, …, un} , then we have and • That is, the matrix for T times the coordinate matrix for x is the coordinate matrix for T(x). Elementary Linear Algebra

  43. Example • Let T : P1 P2 be the transformations defined by T (p(x)) = xp(x). Find the matrix for T with respect to the standard bases, B = {u1, u2} and B = {v1, v2, v3}, where u1 = 1, u2 = x ; v1 = 1, v2 = x , v3 = x2 • Solution: • T(u1) = T(1) = (x)(1) = x and T(u2)= T(x) = (x)(x) = x2 • [T (u1)]B’= [0 1 0]T [T (u2)]B’= [0 0 1]T • Thus, the matrix for T w.r.t. B and B’ is Elementary Linear Algebra

  44. Example • Let T : R2 R3 be the linear transformation defined by • Find the matrix for the transformation T with respect to the bases B = {u1,u2} for R2 and B = {v1,v2,v3} for R3, where • Solution: Elementary Linear Algebra

  45. Example Elementary Linear Algebra

  46. Theorems • Theorem 8.4.1 • If T : RnRmis a linear transformation and if B and B are the standard bases for Rn and Rm, respectively, then [T]B,B= [T] • Theorem 8.4.2 • If T1 : UV and T2 : VW are linear transformations, and if B, B and B are bases for U, V and W, respectively, then [T2 T1]B,B’= [T2 ]B’,B’’[T1 ]B’’,B Elementary Linear Algebra

  47. Theorem 8.4.3 • If T : VV is a linear operator and if B is a basis for V then the following are equivalent • T is one to one • [T]B is invertible • Moreover, when these equivalent conditions hold [T-1]B= [T]B-1 Elementary Linear Algebra

  48. Direction computation T (x) x (3) (1) Multiply by [T]B,B [T (x)]B [x]B (2) Indirect Computation of a Linear Transformation • An indirect procedure to compute a linear transformation: • Compute the coordinate matrix [x]B • Multiply [x]B on the left by [T]B,B to produce [T (x)]B • Reconstruct T (x) from its coordinate matrix [T (x)]B Elementary Linear Algebra

  49. Example • Let T : P2P2 be linear operator defined by T(p(x)) = p(3x – 5), that is, T (co + c1x + c2x2) = co + c1(3x – 5) + c2(3x – 5)2 • Find [T]B with respect to the basis B = {1, x, x2} • Use the indirect procedure to compute T (1 + 2x + 3x2) • Check the result by computing T (1 + 2x + 3x2) • Solution: • Form the formula for T, T(1) = 1, T(x) = 3x – 5, T(x2) = (3x – 5)2 = 9x2 – 30x + 25 • Thus, Elementary Linear Algebra

  50. Direction computation T (x) x (3) (1) Multiply by [T]B,B [T (x)]B [x]B (2) Example • The coordinate matrix relative to B for vector p = 1 + 2x + 3x2 is [p]B = [1 2 3]T. • Thus, [T (1 + 2x + 3x2)]B = [T (p)]B= [T]B [p]B = T (1 + 2x + 3x2) = 66 – 84x + 27x2 • By direction computation: • T (1 + 2x + 3x2) = 1 + 2(3x– 5) + 3(3x– 5)2 = 1 + 6x– 10 + 27x2 – 90x + 75 = 66 – 84x + 27x2 Elementary Linear Algebra

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