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

2. 2015/7/2 Elementary Linear Algebra 2 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

3. 2015/7/2 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 transformation from 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 operator on V.

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 operator on V. 2015/7/2 Elementary Linear Algebra 4

5. 2015/7/2 Elementary Linear Algebra 5 Orthogonal Projections Suppose that W is a finite-dimensional subspace of an inner product space V ; then the orthogonal projection of V onto W is the transformation defined by T (v) = proj W v If S = {w 1, w 2, …, w r } is any orthogonal basis for W, then T (v) is given by the formula T (v) = proj W v =  v, w 1  w 1 +  v, w 2  w 2 + ··· +  v, w r  w r This projection a linear transformation:  T(u + v) = T(u) + T(v)  T(cu) = cT(u)

6. 2015/7/2 Elementary Linear Algebra 6 A Linear Transformation from a Space V to R n Let S = {w 1, w 2, …, w n } be a basis for an n-dimensional vector space V, and let (v) s = (k 1,, k 2,, …, k n ) be the coordinate vector relative to S of a vector v in V; thus v = k 1 w 1 + k 2 w 2 + …+ k n w n Define T : V  R n to be the function that maps v into its coordinate vector relative to S; that is, T (v) = (v) s = (k 1,, k 2,, …, k n ) Then the function T is a linear transformation:  Let u = c 1 w 1 + c 2 w 2 + …+ c n w n and v = d 1 w 1 + d 2 w 2 + …+ d n w n  Check if (u + v) s = (u) s + (v) s and (ku) s = k(u) s

7. 2015/7/2 Elementary Linear Algebra 7 A Linear Transformation from P n to P n+1 Let p = p(x) = c 0 + c 1 x + ··· + c n x n be a polynomial in P n, and define the function T : P n  P n+1 by T (p) = T (p(x)) = xp(x) = c 0 x + c 1 x 2 + ··· + c n x n+1 The function T is a linear transformation:  For any scalar k and any polynomials p 1 and p 2 in P n we have T (p 1 + p 2 ) = T (p 1 (x) + p 2 (x)) = x (p 1 (x) + p 2 (x)) = x p 1 (x) + x p 2 (x) = T (p 1 ) + T (p 2 ) T (k p) = T (k p(x)) = x (k p(x)) = k (x p(x))= k T(p)

8. 2015/7/2 Elementary Linear Algebra 8 A Linear Transformation Using an Inner Product Let V be an inner product space and let v 0 be any fixed vector in V. Let T : V  R be the transformation that maps a vector v into its inner product with v 0 ; that is, T (v) =  v, v 0  From the properties of an inner product  T (u + v) =  u + v, v 0  =  u, v 0  +  v, v 0   T (k u) =  k u, v 0  = k  u, v 0  = kT (u) Thus, T is a linear transformation.

9. 2015/7/2 Elementary Linear Algebra 9 Example Let T:M nn →R be the transformation that maps an n × n matrix into its determinant; that is, T (A) = det (A) If n>1, then this transformation does not satisfy either of the properties required of a linear transformation. For example, we saw Example 1 of Section 2.3 that det (A 1 +A 2 ) ≠ det (A 1 ) + det (A 2 ) in general. Moreover, det (cA) =C n det (A), so det (cA) ≠ c det (A) in general. Thus, T is not linear transformation.

10. 2015/7/2 Elementary Linear Algebra 10 Properties of Linear Transformation If T : V  W is a linear transformation, then for any vectors v 1 and v 2 in V and any scalars c 1 and c 2, we have T (c 1 v 1 + c 2 v 2 ) = T (c 1 v 1 ) + T (c 2 v 2 ) = c 1 T (v 1 ) + c 2 T (v 2 ) More generally, if v 1, v 2, …, v n are vectors in V and c 1, c 2, …, c n are scalars, then T (c 1 v 1 + c 2 v 2 +…+ c n v n ) = c 1 T (v 1 ) + c 2 T (v 2 ) +…+ c n T (v n ) The above equation is sometimes described by saying that linear transformations preserve linear combinations.

11. 2015/7/2 Elementary Linear Algebra 11 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

12. 2015/7/2 Elementary Linear Algebra 12 Finding Linear Transformations from Images of Basis If T : V  W is a linear transformation, and if {v 1, v 2, …, v n } is any basis for V, then the image T (v) of any vector v in V can be calculated from the images T (v 1 ), T (v 2 ), …, T (v n ) of the basis vectors. This can be done by first expressing v as a linear combination of the basis vectors, say v = c 1 v 1 + c 2 v 2 + …+ c n v n and then the transformation becomes T (v) = c 1 T (v 1 ) + c 2 T (v 2 ) + … + c n T (v n ) A linear transformation is completely determined by its images of any basis vectors.

13. 2015/7/2 Elementary Linear Algebra 13 Example Consider the basis S = {v 1, v 2, v 3 } for R 3, where v 1 = (1,1,1), v 2 = (1,1,0), and v 3 = (1,0,0). Let T: R 3  R 2 be the linear transformation such that T (v 1 ) = (1,0), T (v 2 ) = (2,-1), T (v 3 ) = (4,3). Find a formula for T (x 1, x 2, x 3 ); then use this formula to compute T (2, -3, 5).

14. 2015/7/2 Elementary Linear Algebra 14 Composition of T 2 with T 1 Definition  If T 1 : U  V and T 2 : V  W are linear transformations, the composition of T 2 with T 1, denoted by T 2  T 1 (read “T 2 circle T 1 ”), is the function defined by the formula (T 2  T 1 )(u) = T 2 (T 1 (u)) where u is a vector in U. Theorem 8.1.2  If T 1 : U  V and T 2 : V  W are linear transformations, then (T 2  T 1 ) : U  W is also a linear transformation.

15. 2015/7/2 Elementary Linear Algebra 15 Remark The compositions can be defined for more than two linear transformations. For example, if T 1 : U  V and T 2 : V  W,and T 3 : W  Y are linear transformations, then the composition T 3  T 2  T 1 is defined by (T 3  T 2  T 1 )(u) = T 3 (T 2 (T 1 (u)))

16. 2015/7/2 Elementary Linear Algebra 16 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

17. 2015/7/2 Elementary Linear Algebra 17 Kernel and Range Recall:  If A is an m  n matrix, then the nullspace of A consists of all vector x in R n such that Ax = 0.  The column space of A consists of all vectors b in R m for which there is at least one vector x in R n such that Ax = b.  The nullspace of A consists of all vectors in R n that multiplication by A maps into 0. (in terms of matrix transformation)  The column space of A consists of all vectors in R m that are images of at least one vector in R n under multiplication by A. (in terms of matrix transformation)

18. 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). 2015/7/2 Elementary Linear Algebra 18

19. 2015/7/2 Elementary Linear Algebra 19 Examples If T A : R n  R m is multiplication by the m  n matrix A, then the kernel of T A is the nullspace of A and the range of T A 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}.

20. 2015/7/2 Elementary Linear Algebra 20 Example Let T : R 3  R 3 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 R 3 into the xy-plane, the range of T must be some subset of this plane. But every point (x 0,y 0,0) in the xy-plane is the image under T of some point. Thus R(T) is the entire xy-plane.

21. 2015/7/2 Elementary Linear Algebra 21 Example Let T : R 2  R 2 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) = R 2. The only vector that rotates into 0 is 0, so ker(T) = {0}.

22. 2015/7/2 Elementary Linear Algebra 22 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.

23. 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 T A : R n  R m is multiplication by A, then: nullity (T A ) = nullity (A) rank (T A ) = rank (A) 2015/7/2 Elementary Linear Algebra 23

24. 2015/7/2 Elementary Linear Algebra 24 Example Let T A : R 6  R 4 be multiplication by Find the rank and nullity of T A 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 (T A ) = 2 and nullity (T A ) = 4.

25. 2015/7/2 Elementary Linear Algebra 25 Example Let T : R 3  R 3 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.

26. 2015/7/2 Elementary Linear Algebra 26 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.

27. 2015/7/2 Elementary Linear Algebra 27 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

28. 2015/7/2 Elementary Linear Algebra 28 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 T A : R n  R n is multiplication by A, then T A is one-to-one if and only if A is an invertible matrix (Theorem 4.3.1).

29. Example Let T : R 2  R 2 be the linear operator that rotates each vector in the xy-plane through an angle . We showed that ker(T) = {0} and R(T) = R 2.  Thus, rank(T) + nullity(T) = 2 + 0 = 2. 2015/7/2 Elementary Linear Algebra 29

30. 2015/7/2 Elementary Linear Algebra 30 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

31. 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 2015/7/2 Elementary Linear Algebra 31

32. 2015/7/2 Elementary Linear Algebra 32 Example Let T A : R 4  R 4 be multiplication by Determine whether T A is one to one. Solution:  det(A) = 0, since the first two rows of A are proportional  A is not invertible  T A is not one-to-one.

33. 2015/7/2 Elementary Linear Algebra 33 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

34. 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 not onto). For the special case that T : V  V, then the linear transformation is one-to-one and onto. 2015/7/2 Elementary Linear Algebra 34

35. 2015/7/2 Elementary Linear Algebra 35 Example (An Inverse Transformation) Let T : R 3  R 3 be the linear operator defined by the formula T (x 1, x 2, x 3 ) = (3x 1 + x 2, -2x 1 – 4x 2 + 3x 3, 5x 1 + 4 x 2 – 2x 3 ). Determine whether T is one-to-one; if so, find T -1 (x 1,x 2,x 3 ). Solution:

36. 2015/7/2 Elementary Linear Algebra 36 Theorem 8.3.3 If T 1 : U  V and T 2 : V  W are one to one linear transformation then:  T 2  T 1 is one to one  (T 2  T 1 ) -1 = T 1 -1  T 2 -1

37. 2015/7/2 Elementary Linear Algebra 37 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

38. 2015/7/2 Elementary Linear Algebra 38 Matrices of General Linear Transformations Remark:  If V and W are finite-dimensional vector spaces (not necessarily R n and R m ), 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.

39. 2015/7/2 Elementary Linear Algebra 39 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 R n, and coordinate matrix [T(x)] B will be a vector in R m. A vector in V (n-dimensional) A vector in R n A vector in W (m-dimensional) A vector in R m x T (x) [x]B[x]B [T (x)] B T ?

40. 2015/7/2 Elementary Linear Algebra 40 Matrices of Linear Transformations 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 x T (x) [x]B[x]B [T (x)] B T T maps V into W A Multiplication by A maps R n to R m

41. 2015/7/2 Elementary Linear Algebra 41 Matrices of Linear Transformations Let B = {u 1, …, u n } be a basis for the n-dimensional space V and B = {u 1, …, u m } 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 vectors u 1, …, u n. Thus, we need A [u 1 ] B = [T(u 1 )] B, A [u 2 ] B = [T(u 2 )] B, …, A [u n ] B = [T(u n )] B Since [u 1 ] B = e 1, [u 2 ] B = e 2, …, [u n ] B = e n

42. 2015/7/2 Elementary Linear Algebra 42 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 spaceBasis for the domain

43. 2015/7/2 Elementary Linear Algebra 43 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 = {u 1, …, u n }, then we have and That is, the matrix for T times the coordinate matrix for x is the coordinate matrix for T(x).

44. 2015/7/2 Elementary Linear Algebra 44 Example Let T : P 1  P 2 be the transformations defined by T (p(x)) = xp(x). Find the matrix for T with respect to the standard bases, B = {u 1, u 2 } and B = {v 1, v 2, v 3 }, where u 1 = 1, u 2 = x ; v 1 = 1, v 2 = x, v 3 = x 2 Solution:  T(u 1 ) = T(1) = (x)(1) = x and T(u 2 ) = T(x) = (x)(x) = x 2  [T (u 1 )] B ’ = [0 1 0] T [T (u 2 )] B ’ = [0 0 1] T  Thus, the matrix for T w.r.t. B and B ’ is

45. 2015/7/2 Elementary Linear Algebra 45 Example Let T : R 2  R 3 be the linear transformation defined by Find the matrix for the transformation T with respect to the bases B = {u 1,u 2 } for R 2 and B = {v 1,v 2,v 3 } for R 3, where Solution:

46. 2015/7/2 Elementary Linear Algebra 46 Example

47. 2015/7/2 Elementary Linear Algebra 47 Theorems Theorem 8.4.1  If T : R n  R m is a linear transformation and if B and B are the standard bases for R n and R m, respectively, then [T] B,B = [T] Theorem 8.4.2  If T 1 : U  V and T 2 : V  W are linear transformations, and if B, B  and B are bases for U, V and W, respectively, then [T 2  T 1 ] B,B ’ = [T 2 ] B ’,B ’’ [T 1 ] B ’’,B

48. 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 2015/7/2 Elementary Linear Algebra 48

49. 2015/7/2 Elementary Linear Algebra 49 Indirect Computation of a Linear Transformation An indirect procedure to compute a linear transformation: 1)Compute the coordinate matrix [x] B 2)Multiply [x] B on the left by [T] B,B to produce [T (x)] B 3)Reconstruct T (x) from its coordinate matrix [T (x)] B x T (x) [x]B[x]B [T (x)] B Direction computation Multiply by [T] B,B (1) (2) (3)

50. 2015/7/2 Elementary Linear Algebra 50 Example Let T : P 2  P 2 be linear operator defined by T(p(x)) = p(3x – 5), that is, T (c o + c 1 x + c 2 x 2 ) = c o + c 1 (3x – 5) + c 2 (3x – 5) 2  Find [T] B with respect to the basis B = {1, x, x 2 }  Use the indirect procedure to compute T (1 + 2x + 3x 2 )  Check the result by computing T (1 + 2x + 3x 2 ) Solution:  Form the formula for T, T(1) = 1, T(x) = 3x – 5, T(x 2 ) = (3x – 5) 2 = 9x 2 – 30x + 25  Thus,

51. 2015/7/2 Elementary Linear Algebra 51 Example  The coordinate matrix relative to B for vector p = 1 + 2x + 3x 2 is [p] B = [1 2 3] T.  Thus, [T (1 + 2x + 3x 2 )] B = [T (p)] B = [T] B [p] B =  T (1 + 2x + 3x 2 ) = 66 – 84x + 27x 2  By direction computation: T (1 + 2x + 3x 2 ) = 1 + 2(3x – 5) + 3(3x – 5) 2 = 1 + 6x – 10 + 27x 2 – 90x + 75 = 66 – 84x + 27x 2 x T (x) [x]B[x]B [T (x)] B Direction computation Multiply by [T] B,B (1) (2) (3)

52. 2015/7/2 Elementary Linear Algebra 52 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

53. 2015/7/2 Elementary Linear Algebra 53 Similarity The matrix of a linear operator T : V  V depends on the basis selected for V that makes the matrix for T as simple as possible – a diagonal or triangular matrix.

54. 2015/7/2 Elementary Linear Algebra 54 Simple Matrices for Linear Operators Consider the linear operator T : R 2  R 2 defined by and the standard basis B = {e 1, e 2 } for R 2. The matrix for T with respect to this basis is the standard matrix for T; that is, [T] B = [T] = [T(e 1 ) | T(e 2 )]. Since T (e 1 ) = [1 -2] T, T (e 2 ) = [1 4] T, we have However, if u 1 = [1 1] T, u 2 = [1 2] T, then the matrix for T with respect to the basis B = {u 1, u 2 } is the diagonal matrix

55. 2015/7/2 Elementary Linear Algebra 55 Theorem 8.5.1 If B and B are bases for a finite-dimensional vector space V, and if I : V  V is the identity operator, then [I] B,B is the transition matrix from B to B. Remark V I vv Basis = B V [I] B,B is the transition matrix from B to B.

56. 2015/7/2 Elementary Linear Algebra 56 Theorem Theorem 8.5.2  Let T : V  V be a linear operator on a finite-dimensional vector space V, and let B and B be bases for V. Then [T] B = P -1 [T] B P where P is the transition matrix from B to B. Remark: V IIT Basis = B VVV v v T(v)T(v)T(v)T(v) [T] B = [I] B,B [T] B [I] B,B = P -1 [T] B P

57. 2015/7/2 Elementary Linear Algebra 57 Example Let T : R 2  R 2 be defined by Find the matrix T with respect to the standard basis B = {e 1, e 2 } for R 2, then use Theorem 8.5.2 to find the matrix of T with respect to the basis B = {u 1, u 2 }, where u 1 = [1 1] T and u 2 = [1 2] T. Solution:

58. 2015/7/2 Elementary Linear Algebra 58 Definitions Definition  If A and B are square matrices, we say that B is similar to A if there is an invertible matrix P such that B = P -1 AP Definition  A property of square matrices is said to be a similarity invariant or invariant under similarity if that property is shared by any two similar matrices.

59. 2015/7/2 Elementary Linear Algebra 59 Similarity Invariants PropertyDescription DeterminantA and P -1 AP have the same determinant. InvertibilityA is invertible if and only if P -1 AP is invertible. RankA and P -1 AP have the same rank. NullityA and P -1 AP have the same nullity. TraceA and P -1 AP have the same trace. Characteristic polynomialA and P -1 AP have the same characteristic polynomial. EigenvaluesA and P -1 AP have the same eigenvalues Eigenspace dimension If is an eigenvalue of A and P -1 AP then the eigenspace of A corresponding to and the eigenspace of P -1 AP corresponding to have the same dimension.

60. 2015/7/2 Elementary Linear Algebra 60 Determinant of A Linear Operator Two matrices representing the same linear operator T : V  V with respect to different bases are similar. For any two bases B and B we must have det([T] B ) = det([T] B ) Thus we define the determinant of the linear operator T to be det(T) = det([T] B ) where B is any basis for V. Example  Let T : R 2  R 2 be defined by

61. 2015/7/2 Elementary Linear Algebra 61 Eigenvalues of a Linear Operator A scalar is called an eigenvalue of a linear operator T : V  V if there is a nonzero vector x in V such that Tx = x. The vector x is called an eigenvector of T corresponding to. Equivalently, the eigenvectors of T corresponding to are the nonzero vectors in the kernel of I – T. This kernel is called the eigenspace of T corresponding to.

62. Eigenvalues of a Linear Operator If V is a finite-dimensional vector space, and B is any basis for V, then  The eigenvalues of T are the same as the eigenvalues of [T] B.  A vector x is an eigenvector of T corresponding to [T] B if and only if its coordinate matrix [x] B is an eigenvector of [T] B corresponding to. 2015/7/2 Elementary Linear Algebra 62

63. 2015/7/2 Elementary Linear Algebra 63 Example Find the eigenvalues and bases for the eigenvalues of the linear operator T : P 2  P 2 defined by T (a + bx + cx 2 ) = -2c + (a + 2b + c)x + (a + 3c)x 2 Solution:  The eigenvalues of T are  = 1 and  = 2  The eigenvectors of [T] B are:  = 2:  = 1:

64. 2015/7/2 Elementary Linear Algebra 64 Example Let T : R 3  R 3 be the linear operator given by Find a basis for R 3 relative to which the matrix for T is diagonal. Solution:  det(    

65. 2015/7/2 Elementary Linear Algebra 65 Onto Transformations Let V and W be real vector spaces. We say that the linear transformation T : V  W is onto if the range of T is W. An onto transformation is also said to be surjective or to be a surjection. For a surjective mapping, the range and the codomain coincide. If a transformation T : V  W is both one-to-one (also called injective or an injection) and onto, then it is a one-to-one mapping to its range W and so has an inverse T -1 : W  V. A transformation that is one-to-one and onto is also said to be bijective or to be a bijection between V and W.

66. Theorem 8.6.1 Bijective Linear Transformation  Let V and W be finite-dimensional vector spaces. If dim(V)  dim(W), then there can be no bijective linear transformation from V to W. 2015/7/2 Elementary Linear Algebra 66

67. 2015/7/2 Elementary Linear Algebra 67 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

68. 2015/7/2 Elementary Linear Algebra 68 Isomorphisms Definition  An isomorphism between V and W is a bijective linear transformation from V to W.

69. Isomorphisms Theorem 8.6.2 (Isomorphism Theorem)  Let V be a finite-dimensional real vector space. If dim(V) = n, then there is an isomorphism from V to R n. Example  The vector space P 3 is isomorphic to R 4, because the transformation T(a + bx + cx 2 + dx 3 ) = (a,b,c,d) is one-to-one, onto, and linear. 2015/7/2 Elementary Linear Algebra 69

70. 2015/7/2 Elementary Linear Algebra 70 Isomorphisms between Vector Spaces Theorem 8.6.3 (Isomorphism of Finite-Dimensional Vector Spaces)  Let V and W be finite-dimensional vector spaces. If dim(V) = dim(W), then V and W are isomorphic.

71. Example An Isomorphism between P 3 and M 22  Because dim(P 3 ) = 4 and dim(M 22 ) = 4, these spaces are isomorphic.  We can find an isomorphism T : P 3  M 22 :  This is one-to-one and onto linear transformation, so it is an isomorphism between P 3 and M 22. 2015/7/2 Elementary Linear Algebra 71