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Elementary Linear Algebra Anton & Rorres, 9th 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 2017/5/6 Elementary Linear Algebra 2 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. 2017/5/6 Elementary Linear Algebra 3 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. 2017/5/6 Elementary Linear Algebra 4 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) = 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) 2017/5/6 Elementary Linear Algebra 5 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 Define T : 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: 2017/5/6 Let u = c1w1 + c2w2 + …+ cn wn and v = d1w1 + d2w2 + …+ dn wn Check if (u + v)s = (u)s + (v)s and (ku)s = k(u)s Elementary Linear Algebra 6 A Linear Transformation from Pn to Pn+1 Let p = p(x) = c0 + c1x + ··· + cnx n be a polynomial in Pn , and define the function T : Pn Pn+1 by 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 2017/5/6 T (p1 + p2) = T (p1(x) + p2 (x)) = x (p1(x) + p2 (x)) = x p1(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 7 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. 2017/5/6 Elementary Linear Algebra 8 Properties of Linear Transformation If T : V W is a linear transformation, then for any vectors v1 and v2 in V and any scalars c1 and 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. 2017/5/6 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 2017/5/6 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. 2017/5/6 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 R2 be 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). 2017/5/6 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 2017/5/6 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))) 2017/5/6 Elementary Linear Algebra 15 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism 2017/5/6 Elementary Linear Algebra 16 Kernel and Range Recall: If A is an mn 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) m The column space of A consists of all vectors in R that are images of at least one vector in Rn under multiplication by A. (in terms of matrix transformation) 2017/5/6 Elementary Linear Algebra 17 Kernel and Range Definition 2017/5/6 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 Rm is multiplication by the mn 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}. 2017/5/6 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. 2017/5/6 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}. 2017/5/6 Elementary Linear Algebra 21 Properties of Kernel and Range Theorem 8.2.1 2017/5/6 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 mn matrix and TA : Rn Rm is multiplication by A, then: nullity (TA) = nullity (A) rank (TA) = rank (A) 2017/5/6 Elementary Linear Algebra 23 Example 1 0 0 0 Let TA : R6 R4 be multiplication by 5 3 1 2 0 4 3 7 2 0 1 4 A 2 5 2 4 6 1 4 9 2 4 4 7 0 4 28 37 13 1 2 12 16 5 0 0 0 0 0 0 0 0 0 0 x1 4 28 37 13 x 2 12 16 5 2 x3 1 0 0 0 r s t u x4 0 1 0 0 x5 0 0 1 0 0 0 0 1 x6 Find 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. 2017/5/6 Elementary Linear Algebra 24 Example Let T : R3 R3 be the orthogonal projection on the xyplane. 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. 2017/5/6 Elementary Linear Algebra 25 Dimension Theorem for Linear Transformations Theorem 8.2.3 If T : V W is a linear transformation from an ndimensional 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. 2017/5/6 Elementary Linear Algebra 26 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism 2017/5/6 Elementary Linear Algebra 27 One-to-One Linear Transformation A linear transformation T : V W is said to be oneto-one if T maps distinct vectors in V into distinct vectors in W. Examples 2017/5/6 If A is an nn matrix and TA : Rn Rn is multiplication by A, then TA is 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. 2017/5/6 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 2017/5/6 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 2017/5/6 Elementary Linear Algebra 31 Example Let TA : R4 R4 be multiplication by 1 2 A 3 1 3 2 4 6 4 8 9 1 5 1 4 8 Determine 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. 2017/5/6 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 2017/5/6 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 not onto). For the special case that T : V V, then the linear transformation is one-to-one and onto. 2017/5/6 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 + 4 x2 – 2x3). Determine whether T is one-to-one; if so, find T -1(x1,x2,x3) . Solution: 3 1 0 [T ] 2 4 3 5 4 2 4 2 3 [T ]1 11 6 9 12 7 10 x1 x1 4 2 3 x1 4 x1 2 x2 3x3 T 1 x2 [T 1 ] x2 11 6 9 x2 11x1 6 x2 9 x3 x x3 12 7 10 x3 12 x1 7 x2 10 x3 3 T 1 ( x1 , x2 , x3 ) (4 x1 2 x2 3x3 ,11x1 6 x2 9 x3 ,12 x1 7 x2 10 x3 ) 2017/5/6 Elementary Linear Algebra 35 Theorem 8.3.3 If T1 : U V and T2 : V W are one to one linear transformation then: 2017/5/6 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 2017/5/6 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. 2017/5/6 Elementary Linear Algebra 38 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)] m B will be a vector in R . T A vector in V (n-dimensional) x A vector in Rn [x]B 2017/5/6 T (x) ? [T (x)]B Elementary Linear Algebra A vector in W (m-dimensional) A vector in Rm 39 Matrices of Linear Transformations T maps V into W 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 T (x) x [x]B A [T (x)]B Multiplication by A maps Rn to Rm 2017/5/6 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. a11 a12 a1n Consider an mn matrix a a a 21 22 2 n A am1 am 2 amn 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 u1, …, 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 2017/5/6 Elementary Linear Algebra 41 Matrices of Linear Transformations We have Thus, A [[T (u1 )]B' | [T (u2 )]B' | | [T (un )]B',] which is the matrix for T w.r.t. the bases B and B, and denoted by the symbol [T]B,B That is, a11 a1n a a [T (u1 )]B ' A[u1 ]B A e1 21 , ...... , T [(u1 )] B ' A[u n ]B A e n 2 n a m1 amn [T ]B ', B [[T (u1 )]B ' | [T (u 2 )]B ' | | [T (u n )] B ' ] and [T ]B ', B [x]B [T (x)] B ' Basis for the image space 2017/5/6 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 [T ]B [[T (u1 )]B | [T (u 2 )]B | | [T (u n )]B ] and [T ]B [x]B [T (x)]B That is, the matrix for T times the coordinate matrix for x is the coordinate matrix for T(x). 2017/5/6 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 [T (u1)]B’ = [0 1 0] [T (u2)]B’ = [0 0 1]T Thus, the matrix for T w.r.t. B and B’ is 0 0 [T ]B ', B [[T (u1 )]B ' | [T (u 2 )]B ' ] 1 0 0 1 2017/5/6 44 Elementary Linear Algebra Example Let T : R2 R3 be the linear transformation defined by x2 x1 T 5 x1 13x2 x2 7 x 16 x 1 2 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 1 1 0 3 5 u1 , u 2 , v1 0 , v 2 2 , v 3 1 1 2 1 2 2 1 2 Solution: T (u ) 2 v 2 v , T (u ) 1 3v v v 1 1 3 2 1 2 3 5 3 2017/5/6 Elementary Linear Algebra 45 Example 1 2 T (u1 ) 2 v1 2 v 3 , T (u 2 ) 1 3v1 v 2 v 3 5 3 1 3 [T (u1 )]B ' 0 , [T (u 2 )]B ' 1 2 1 3 1 [T ]B ', B [[T (u1 )]B ' | [T (u 2 )]B ' ] 0 1 2 1 2017/5/6 Elementary Linear Algebra 46 Theorems Theorem 8.4.1 If T : Rn Rm is 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 2017/5/6 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 2017/5/6 Elementary Linear Algebra 48 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 Direction computation (3) (1) [x]B 2017/5/6 T (x) Multiply by [T]B,B (2) Elementary Linear Algebra [T (x)]B 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 2 Find [T]B with respect to the basis B = {1, x, x } 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, 1 5 25 [T ]B 0 3 30 0 0 9 2017/5/6 Elementary Linear Algebra 50 Example 1 5 25 [T ]B 0 3 30 0 0 9 The coordinate matrix relative to B for vector p = 1 + 2x + 3x2 is [p]B = [1 2 3]T. 1 5 25 1 66 Thus, [T (1 + 2x + 3x2)]B = [T (p)]B = [T]B [p]B = 0 3 30 2 84 0 0 9 3 27 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 x Direction computation (3) (1) Multiply by [T]B,B [x]B 2017/5/6 Elementary Linear Algebra T (x) (2) [T (x)]B 51 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism 2017/5/6 Elementary Linear Algebra 52 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. 2017/5/6 Elementary Linear Algebra 53 Simple Matrices for Linear Operators Consider the linear operator T : R2 R2 defined by T x1 x1 x2 x 2 x 4 x 2 1 2 2 and the standard basis B = {e1, e2} for R . The matrix for T with respect to this basis is the standard matrix for T; that is, [T]B = [T] = [T(e1) | T(e2)]. Since T (e1) = [1 -2]T, T (e2) = [1 4]T, we have [T ]B 1 1 2 4 However, if u1 = [1 1]T, u2 = [1 2]T, then the matrix for T with respect to the basis B = {u1, u2} is the diagonal matrix 2 0 [T ]B ' 0 3 2017/5/6 Elementary Linear Algebra 54 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 I V v Basis = B V v Basis = B [I]B,B is the transition matrix from B to B. 2017/5/6 Elementary Linear Algebra 55 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: I V v Basis = B T V v Basis = B V I T(v) Basis = B V T(v) Basis = B [T]B = [I]B,B[T]B[I]B,B = P-1 [T]B P 2017/5/6 Elementary Linear Algebra 56 Example Let T : R2 R2 be defined by x x x T 1 1 2 x2 2 x1 4 x2 Find the matrix T with respect to the standard basis B = {e1, e2} for R2, then use Theorem 8.5.2 to find the matrix of T with respect to the basis B = {u1, u2}, where u1 = [1 1]T and u2 = [1 2]T. 1 1 [T ]B 2 4 Solution: P [ I ]B.B ' [[u1 ' ]B | [u 2 ' ]B ] 2 1 P 1 1 1 2 1 1 1 1 1 2 0 P 1 T B P 1 1 2 4 1 2 0 3 1 1 P 1 2 T B ' 2017/5/6 Elementary Linear Algebra 57 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-1AP Definition 2017/5/6 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. Elementary Linear Algebra 58 Similarity Invariants Property Description Determinant A and P-1AP have the same determinant. Invertibility A is invertible if and only if P-1AP is invertible. Rank A and P-1AP have the same rank. Nullity A and P-1AP have the same nullity. Trace A and P-1AP have the same trace. Characteristic polynomial A and P-1AP have the same characteristic polynomial. Eigenvalues A and P-1AP have the same eigenvalues Eigenspace dimension If is an eigenvalue of A and P-1AP then the eigenspace of A corresponding to and the eigenspace of P-1AP corresponding to have the same dimension. 2017/5/6 Elementary Linear Algebra 59 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 x1 x1 x2 2 2 Let T : R R be defined by T x 2 x 4 x 1 2 2 1 1 [T ]B 2 4 2 0 T B ' 0 3 2017/5/6 det(T ) 6 det(T ) 6 Elementary Linear Algebra 60 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 . 2017/5/6 Elementary Linear Algebra 61 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 . 2017/5/6 Elementary Linear Algebra 62 Example Find the eigenvalues and bases for the eigenvalues of the linear operator T : P2 P2 defined by T (a + bx + cx2) = -2c + (a + 2b + c)x + (a + 3c)x2 Solution: 0 0 2 1 The eigenvalues of T are = 1 and = 2 T B 1 2 1 0 3 The eigenvectors of [T]B are: 2 1 0 = 2: = 1: u1 0 , u 2 1 1 0 2017/5/6 Elementary Linear Algebra u 3 1 1 63 Example Let T : R3 R3 be the linear operator given by x1 2 x3 T x2 x1 2 x2 x3 x x 3x 3 3 1 Find a basis for R3 relative to which the matrix for T is diagonal. Solution: det(I A) 3 52 8 4 (2)(2)(1) 2 0 0 [T ]B ' 0 2 0 0 0 1 2017/5/6 Elementary Linear Algebra 64 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. 2017/5/6 Elementary Linear Algebra 65 Theorem 8.6.1 Bijective Linear Transformation 2017/5/6 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. Elementary Linear Algebra 66 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism 2017/5/6 Elementary Linear Algebra 67 Isomorphisms Definition An isomorphism between V and W is a bijective linear transformation from V to W. 2017/5/6 Elementary Linear Algebra 68 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 Rn. Example 2017/5/6 The vector space P3 is isomorphic to R4, because the transformation T(a + bx + cx2 + dx3) = (a,b,c,d) is one-to-one, onto, and linear. Elementary Linear Algebra 69 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. 2017/5/6 Elementary Linear Algebra 70 Example An Isomorphism between P3 and M22 Because dim(P3) = 4 and dim(M22) = 4, these spaces are isomorphic. We can find an isomorphism T : P3 M22: 1 0 0 1 0 0 0 0 2 3 T (1) T ( x) T (x ) T (x ) 0 0 0 0 1 0 0 1 2017/5/6 This is one-to-one and onto linear transformation, so it is an isomorphism between P3 and M22. Elementary Linear Algebra 71