EXERCISES IN LINEAR ALGEBRA. 1. Matrix operations

EXERCISES IN LINEAR ALGEBRA 1 Matrix operations (1) Put D = diag(d 1, d 2,, d n ) Let A = (a ij ) be an n n matrix Find DA and AD When is D invertible? (2) An n n matrix A = (a ij ) is called upper triangular if a ij = 0 whenever i > j Prove that product of upper triangular matrices is upper triangular (3) Let A be an m n matrix and B be an n p matrix Let A i denote the i th row of A and A j denote the j th column of A Show that AB = (AB 1, AB 2,, AB p ) and AB = (A 1 B, A 2 B,, A m B) t (4) Prove that a matrix that has a zero row or a zero column is not invertible (5) A square matrix A is called nilpotent if A k = 0 for some positive integer k Show that if A is nilpotent then I + A is invertible (6) Find infinitely many matrices B such that BA = I 2 where A = 2 3 1 2 2 5 Show that there is no matrix C such that AC = I 3 (7) Let A and B be square matrices Let tr(a) denote the trace of A which is the sum of its diagonal entries Show that for two n n matrices A and B, tr(a + B) = tr(a) + tr(b) and tr(ab) = tr(ba) Show that if B is invertible then tr(a) = tr(bab 1 ) (8) Show that the equation AB BA = I has no solutions in R n n (9) Show that for any matrix A, AA t is symmetric Show that every square matrix is uniquely a sum of a symmetric and skew-symmetric matrix (10) Show that every matrix in C n n is uniquely a sum of a Hermitian and skew-hermitian matrix (11) Show that inverse of an invertible symmetric matrix is also symmetric (12) Consider a system of linear equations Ax = b where A R m n, x = (x 1, x 2,, x n ) t and b R m (a) Show that if Ax = b has more than one solution then it has infinitely many (b) Prove that if there is a complex solution then there is a real solution (13) Find all 2 2 matrices A such that A 2 = I 1

2 (14) Find all 2 2 matrices A such that A 2 = 0 (15) Show that if A 3 A + I = 0 then A is invertible 2 Vector spaces, subspaces, basis and dimension (16) Determine which of the following subsets of R n are subspaces? (a) V 1 = {(x 1, x 2,, x n ) x 1 = 1} (b) V 2 = {(x 1, x 2,, x n ) x 1 = 0} (c) V 3 = {(x 1, x 2,, x n ) n i=1 x iy i = 0} Here (y 1, y 2,, y n ) R n } is a fixed vector (17) Let V = C(R) = {f : R R f is continuous} Determine which of the following are subspaces of C(R) (a) V 4 = {f V f(1/2) is a rational number} (b) V 5 = {f V 1 f(t)dt = 0} 0 (c) V 5 = {f V a d2 f dt 2 + b df + cf = 0} Here a, b, c R are fixed dt (18) Find a basis of the subspace of R n of the solutions of the equation x 1 +x 2 + +x n = 0 (19) Show that the vector space F x] of all polynomials over a field F is not finitely generated (20) Is the vector space C(R) finite dimensional? (21) Show that the set {1, (x a), (x a) 2, (x a) 3,, (x a) n } for a fixed a F is a basis of the vector space P n (F ) of all polynomials with coefficients in F of degree atmost n (22) Let u 1, u 2, u n be linearly independent vectors in a vector space V Show that any vector in L(u 1, u 2,, u n ) is unique linear combination of u 1, u 2,, u n (23) Describe all the subspaces of R 3 (24) Let U and V be subspaces of a vector space W Suppose that U V = (0) and dim W = dim U + dim V Show that any w W there exist unique vectors u U and v V such that w = u + v (25) What is the dimension of the Q-vector space R? (26) Determine whether (1, 1, 1) L{(1, 3, 4), (4, 0, 1), (3, 1, 2)} (27) Prove that every subspace W of a finitely generated vector space V is finitely generated Prove that dim W dim V with equality if and only if V = W (28) Let F be a field with two elements Let V be a two dimensional vector space over F Count the number of elements of V, the number of subspaces of V and the number of different bases (29) Let S and T be two dimensional subspaces of R 3 Show that dim(s T ) 1 (30) Find bases of the following vector spaces: (a) the vector space of all n n real upper triangular matrices, (b) the vector space of all real n n symmetric matrices, (c)

the vector space of all real n n skew-symmetric matrices and (d) the vector space of all homogeneous polynomials of degree d in n variables together with the zero polynomial 3 Systems of linear equations, rank of a matrix 3 (31) Test for solvability of the following systems of equations, and if solvable, find all the solutions (a) (b) x 1 + x 2 + x 3 = 8 x 1 + x 2 + x 4 = 1 x 1 + x 3 + x 4 = 14 x 2 + x 3 + x 4 = 14 x 1 + 2x 2 + 4x 3 = 1 2x 1 + x 2 + 5x 3 = 0 3x 1 x 2 + 5x 3 = 0 (32) For what vaules of a does the following system of equations have a solution? 3x 1 x 2 + ax 3 = 1 3x 1 x 2 + x 3 = 5 (33) Prove that a system of m homogeneous linear equations in n > m unknowns always has a nontrivial solution (34) Show that a system of homogeneous linear equations in n unknowns has a nontrivial solution if and only if the coefficient matrix has rank less than n (35) Find a basis of the solution space of the system 3x 1 x 2 + x 4 = 0 x 1 + x 2 + x 3 + x 4 = 0 (36) Find a point in R 3 where the line joining the points (1, 1, 0) and ( 2, 1, 1) pierces the plane 3x 1 x 2 + x 3 1 = 0 (37) Using row and column operations find the rank of the matrix 1 2 3 1 2 3 4 8 12 1 1 5

4 (38) Find the rank of an upper triangular matrix in terms of the diagonal entries (39) Let A be an m n matrix and B be an n r matrix (a) Show that the columns of AB are linear combinations of the columns A Hence prove that rank(ab) rank(a) (b) Using (a) and the fact that rank of a matrix and its transpose are equal, prove that rank(ab) rank(b) (40) Let A be an n n matrix such that rank(a) = rank(a 2 ) Find all the vectors in the column space of A which are solutions to Ax = 0 4 Linear Transformations (41) Let F be a field and F n denote the vector space F n 1 Let T : F 2 F 2 be the linear transformation T ((x, y) t ) = ( 3x + y, x y) t Let U : F 2 F 2 be the linear transformation U((x, y) t ) = (x + y, x) t Describe the linear transformations UT, T U and T 2 + U Is T U = UT? (42) Test whether the linear transformations T : R 2 R 2, T ((x 1, x 2 ) t ) = (y 1, y 2 ) t defined below are one-to-one (a) y 1 = 3x 1 x 2, y 2 = x 1 + x 2 (b) y 1 = x 1 + 2x 2 + x 3, y 2 = x 1 + x 2, y 3 = x 2 + x 3 (43) Let I : Rx] Rx] be the linear map I(f(x)) = x f(x)dx Let D : Rx] Rx] 0 be the linear map D(f(x)) = f (x) Show that DI = I but neither D nor I are isomorphisms Is D (resp I) one-to-one or onto? Find the ranks and nullities of D and I (44) Let S, T : R 2 R 2 be the linear maps defined by the equations S(u 1 ) = u 1 u 2, S(u 2 ) = u 1 and T (u 1 ) = u 2, T (u 2 ) = u 1, where B = {u 1, u 2 } is a basis of R 2 Let C = {w 1 = 3u 1 u 2, w 2 = u 1 + u 2 } Show that C is a basis of R 2 Find the matrices M B B (S), M B B (T ), M C C (S), M C C (S) Find invertible matrices X in each case such that X 1 AX = A where A is the matrix of the transformation with respect to the old basis and A is the matrix of the transformation with respect to the new basis (45) Let B = {u 1, u 2 } be a basis of R 2 Let S and T be the linear maps defined by the equations S(u 1 ) = u 1 + u 2, S(u 2 ) = u 1 u 2 and T (u 1 ) = u 1 u 2, T (u 2 ) = 2u 2 Find the rank and nullity of S and T Which of these linear maps are invertible? (46) Let V be a vector space of dimension n and T : V V be a linear map Let A be the matrix of T with respect to any basis of V Show that rank(t ) = rank(a)

(47) Let V be an n-dimensional vector space Let T : V V be a linear transformation such that the nullspace and the range of T are same Show that n is even Give an example of such a map for n = 2 (48) Let T be the linear operator on R 3 defined by the equations: T ((x 1, x 2, x 3 ) t ) = (3x 1, x 1 x 2, 2x 1 + x 2 + x 3 ) t Is T invertible? If so, find a formula for T 1 (49) Let V = C 2 2 be the vector space of 2 2 complex matrices Let ] 1 1 B = 4 4 Define T : V V by T (A) = BA Find rank of T Describe T 2 (50) Let V be vector space with dim V = n and T : V V be a linear map such that rank T 2 = rank T Show that N(T ) T (V ) = (0) Give an example of such a map (51) Let T be a linear operator on a finite-dimensional vector space V Suppose that U is a linear operator on V such that T U = I Prove that T is invertible and U = T 1 (52) Let W be the real vector space all 2 2 complex Hermitian matrices Show that the map ] (x, y, z, t) t t + x y + iz y iz t x is an isomorphism of R 4 onto W (53) Let V and W be finite dimensional vector spaces over a field F Show that V is isomorphic to W if and only if dim V = dim W (54) Show that every matrix A F m n of rank one has the form A = uv t where u F m 1 and v F n 1 (55) Let V be the infinite-dimensional real vector space of all real sequences Let R : V V be the right shift operator R((a 1, a 2, )) = (0, a 1, a 2, ) and L : V V denote the left shift operator defined by L((a 1, a 2, )) = (a 2, a 3, ) Show that R is one-to-one but not onto and L is onto but not one-to-one (56) Let V be an n-dimensional vector space over a field F Let B = {u 1, u 2,, u n } be a basis of V Define the linear transformation T : V V by T (u i ) = u i+1 for i = 1, 2,, n 1 and T (u n ) = 0 (a) Find the matrix A = MB B (T ) (b) Show that T n 1 0 but T n = 0 (c) Let S be any linear operator on V such that S n = 0 but S n 1 0 Prove that there is a basis C of V such that MC C (S) = A (d) Let M, N F n n and M n = N n = 0 but M n 1 0 and N n 1 0 Show that M and N are similar matrices 5

6 (57) Let V be a two-dimensional vector space over a field F Let T be a linear operator on V so that the matrix of T with respect to a basis B of V is ] a b c d Show that T 2 (a + d)t + (ad bc)i = 0 (58) Let T : V W be a linear map of vector spaces Suppose V is infinite-dimensional Prove that at least one of N(T ) or T (V ) is infinite-dimensional (59) Let V be the vector space of all real functions continuous on a, b] Define T : V V by the equation T (f(x)) = b a f(t) sin(x t)dt for a x b Find the rank and nullity of T (60) Let V denote the vector space of all real functions continuous on the interval π, π] Let S denote the subspace of V consisting of all f satisfying π π f(t)dt = 0, π π f(t) cos tdt = 0, π π f(t) sin tdt = 0 Prove that S contains the functions f(x) = sin nx and f(x) = cos nx for all n = 2, 3, Show that S is infinite-dimensional 5 Inner product spaces (61) Let V be the subspace spanned by the vectors u 1 = ( 1, 1, 1, 1), u 2 = (1, 1, 1, 1), u 3 = (1, 1, 1, 1) in R 4 Find an orthonormal basis of V by Gram-Schmidt process (62) Find an orthonormal basis of the vector space V = P 3 (R) of all real polynomials of degree atmost 3 with the inner product < f, g >= 1 f(t)g(t)dt Take {1, x, 0 x2, x 3 } as a basis of V (63) Let V = C π, π] be the vector space of all continuous real valued functions defined on the interval π, π] Then V is an inner product space with the inner product < f, g >= π f(t)g(t)dt Show that the functions 1, sin nx, cos nx, n = 1, 2, form π an orthogonal set (64) Two vector spaces V and W with inner products < v 1, v 2 > and w 1, w 2 ] respectively, are said to be isometric if there is an isomorphism T : V W such that T (v 1 ), T (v 2 )] =< v 1, v 2 > for all v 1, v 2 V Such a T is called an isometry Let V be a finite-dimensional inner product space over a field F with inner product < u, v > Let B = {v 1, v 2,, v n } be an orthonormal basis of V Let T : V F n be the linear map T (v) = M B (v) Consider F n with standard inner product Show that T is an isometry

(65) Let V be a finite-dimensional inner product space Let W be a subspace Show that the set W = {v V < v, w >= 0 for all w W } is a subspace of V and dim W + dim W = dim V (66) Show that every subspace of C n with standard inner product is the subspace of all solutions to a system of homogeneous linear equations (67) Let A = (a ij ) R 2 2 For u, v R 2 define f A (u, v) = v t Au Show that f A is an inner product on R 2 if and only if A = A t, a 11 > 0, a 22 > 0 and det A > 0 (68) Let V = C n n with the inner product < A, B >= tr(ab ) Let D be the subspace of diagonal matrices Find D (69) Let W be a finite-dimensional subspace of an inner product space V Let E be the orthogonal projection of V onto W Prove that < Eu, v >=< u, Eu > for all u, v V (70) Let V be a finite-dimensional inner product space and let B = {u 1, u 2,, u n } be a basis of V Let T : V V be a linear map Put MB B(T ) = (a ij) Show that a ij =< T u j, u i > (71) Let A be a symmetric n n real matrix Let u, v V = R n 1 \ {0} and λ, µ R such that Au = λu and Av = µv Show that u v In other words, eigenvectors for distinct eigenvalues of symmetric real matrices are orthogonal (72) Let B = {u 1, u 2,, u n } be an orthonormal set of vectors in an inner product space V Show that for any v V, n i=1 v, u i 2 v 2 and equality holds if and only if v L(B) (73) Let p, q, r Z Show that the vectors (p, q, r) t, (q, r, p) t, (r, p, q) t R 3 are mutually orthogonal if and only if pq + qr + rp = 0 Show that in this case, the length of each of these vectors in p + q + r (74) Let U, V be subspaces of an inner product space W Let dim U < dim V Show that there is a nonzero vector in V lying in U (75) Without using Gram-Schmidt orthogonalization process, find the orthogonal projection of (1, 2, 2, 9) t R 4 in the column space of the matrix 7 A = 2 1 1 2 1 0 2 1

8 6 Determinants (76) Find the inverses of the following matrices by Gauss-Jordan method and the adjoint formula: ] 3 1 0 a b, ad bc 0 1 2 1 c d 0 1 2 (77) Find the ranks of the following matrices by using determinants: ] 1 0 1 2 1 2 3 4, 1 1 3 0 1 2 1 0 1 2 4 1 (78) Show that the equation of line through the distinct points (a, b) and (c, d) in R 2 is given by x y 1 a b 1 = 0 c d 1 (79) Show that the equation of the plane in R 3 passing through three non-collinear points (a, b, c); (d, e, f); (g, h, k) is given by x y z 1 a b c 1 d e f 1 g h k 1 = 0 (80) Show that the area of the triangle with vertices (a, b); (c, d); (e, f) in the plane is given by the absolute value of 1 a b 1 c d 1 6 e f 1 (81) Show that the volume of the tetrahedron with vertices (a 1, a 2, a 3 ), (b 1, b 2, b 3 ), (c 1, c 2, c 3 ), (d 1, d 2, d 3 ) is given by the absolute value of 1 6 a 1 a 2 a 3 1 b 1 b 2 b 3 1 c 1 c 2 c 3 1 d 1 d 2 d 3 1

(82) Prove the following formula for the van der Monde determinant: 1 a 1 a 2 1 a n 1 1 1 a V n = 2 a 2 2 a n 1 2 = j a i ) i<j(a 1 a n a 2 n a n 1 n Hint: Use induction on n Multiply each column by a 1 and subtract it from the next column on the right, starting from the right hand side Prove that V n = (a n a 1 )(a n 1 a 1 ) (a 2 a 1 )V n 1 ] (83) If A C n n is a skew-symmetric matrix where n is odd, then show that det A = 0 (84) Let A be an orthogonal matrix, that is, AA t = I Show that for such a matrix, det A = ±1 Give an example of an orthogonal matrix with determinant 1 (85) A complex n n matrix is called unitary if AA = I Here A denotes the conjugate transpose of A If A is unitary, show that det A = 1 (86) Let V = F n n Let B V Define T B : V V by T B (A) = AB BA Show that det T B = 0 (87) Let A, B F n n where A is invertible Show that there are atmost n scalars for which ca + B is not invertible (88) Let V = F n n and B V Define L B, R B : V V by L B (A) = BA and R B (A) = AB for all A V Show that det R B = det L B = (det B) n (89) Let V = F 1 n and let T : V V be a linear operator Define f(u 1, u 2,, u n ) = det(t u 1, T u 2,, T u n ) (1) Show that f is multilinear and alternating (2) Let B be any ordered basis of V and A = T ] B Show that det A = det T = f(e 1, e 2,, e n ) (90) Let B V = C n n Define the linear operator M B : V V by M B (A) = BAB Show that det M B = det B 2n 9 7 Diagonalization of matrices and operators Let V be an n-dimensional vector space over a field F and T be a linear operator on V in the following problems unless stated otherwise Eigenvalues and eigenvectors (91) Show that similar matrices have same characteristic polynomials and hence have same eigenvalues and traces (92) Find the eigenvalues and eigenspaces of the following matrices and determine if they are diagonalizable: 4 5 1 2 ], 1 0 1 2 ]

10 (93) Let f(x) F x] Show that α is an eigenvalue of T if and only if f(α) is an eigenvalue of f(t ) (94) Let A, B F n n Prove that if I AB is invertible then I BA is invertible and (I BA) 1 = I + B(I AB) 1 A (95) Use the result of the above exercise to show that AB and BA have the same characteristic polynomials (96) Let T be an invertible linear operator on a vector space V Show that λ is an eigenvalue of T if and only if λ 0 and λ 1 is an eigenvalue of T 1 (97) Let N C 2 2 and N 2 = 0 Prove that either N = 0 or N is similar over C to 0 0 1 0 (98) Let A C 2 2 Show that A is similar over C to a matrix of one of the two types: ] ] a 0 a 0, 0 b 1 a (99) Let V denote the vector space of all continuous real valued functions defined on R Let T be the linear operator T (f(x)) = x f(t)dt Show that T has no eigenvalues 0 (100) Let A be an n n diagonal matrix with characteristic polynomial C A (x) = (x c 1 ) d 1 (x c 2 ) d 2 (x c k ) d k ] where c 1, c 2,, c k are distinct Let V be the vector space of all n n matrices B such that AB = BA Prove that dim V = d 2 1 + d 2 2 + + d 2 k (101) Find the minimal polynomials of ] 2 0, 3 1 Minimal polynomials 0 1 0 0 0 1 1 0 0, 0 1 3 0 0 2 0 0 0 (102) Find an n n nilpotent matrix with minimal polynomial x 2 (103) Show that the following matrices have the same minimal polynomials: 1 0 0 0 2 0 0 0 0 1 0 0 0 0 2 0, 0 2 0 0 0 0 2 0 0 0 0 1 0 0 0 1 (104) Prove that a linear operator T defined on a finite dimensional vector space V is invertible if and only if its minimal polynomial has a nonzero constant term Describe how to find T 1 from its minimal polynomial (105) Let T be a nilpotent operator on V Show that T n = 0

(106) Let V = F n n Let A V be a fixed matrix Let T be the linear operator on V defined by T (B) = AB Show that T and A have same minimal polynomial (107) Let m(x) be the minimal polynomial of T and f(x) F x] Let d(x) be the greatest common divisor of m(x) and f(x) Show that the nullspaces of f(t ) and d(t ) are equal (108) Show that the matrix Diagonalization 0 0 1 1 0 0 0 1 0 is similar to a diagonal matrix in C 3 3 but not in R 3 3 (109) Let T have n distinct eigenvalues Show that T is diagonalizable (110) Show that every matrix A such that A 2 = A is diagonalizable (111) Show that the orthogonal projection operators are diagonalizable (112) When is a nilpotent operator diagonalizable? (113) Show that the differentiation operator defined on the space of real polynomials of degree atmost n 1 is not diagonalizable (114) Let T : R 4 R 4 be the linear operator induced by the matrix 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c 0 Find necessary and sufficient conditions on a, b, c so that T is diagonalizable (115) Show that a 2 2 real symmetric matrix is diagonalizable 11 8 Projections and invariant direct sums Projections (116) Find a projection E : R 2 R 2 so that E(R 2 ) is the subspace spanned by (1, 1) t and N(E) is spanned by (1, 2) t (117) Let E 1 and E 2 be projections onto independent subspaces of a vector space V Is E 1 + E 2 a projection? (118) Is it true that a diagonalizable operator with only eigenvalues 0 and 1 is a projection? (119) Let E : V V be a projection Show that I E is a projection along E(V ) onto N(E)

12 (120) Let V be a real vector space and let E : V V be a projection Prove that I + E is invertible and find its inverse (121) Let F be a subfield of C Let E 1, E 2,, E k be projections of an n-dimensional F - vector space V such that E 1 + E 2 + + E k = I Prove that E i E j = 0 for i j Hint: Use the fact that the trace of projection is its rank] Invariant subspaces and direct sums (122) Let E be a projection of V and T L(V, V ) Prove that E(V ) is invariant under T if and only if ET E = T E Prove that E(V ) and N(E) are invariant under T if and only if T E = ET (123) Let T : R 2 R 2 be the linear operator induced by the matrix 2 1 0 2 Let W 1 = L(e 1 ) Prove that W 1 is invariant under T Show that there is no subspace W 2 of R 2 that is invariant under T and R 2 = W 1 W 2 (124) Let T be a linear operator on V Suppose V = W 1 W 2 W k where each W i is invariant under T Let T i be the restriction of T on W i (a) Prove that det T = det T 1 det T 2 det T k (b) Show that C T (x) = C T1 (x)c T2 (x) C Tk (x) (c) Show that the minimal polynomial of T is the least common multiple of the minimal polynomials of T 1, T 2,, T k (125) Let T be the linear operator on V = R 3 induced by the matrix A = ] 5 6 6 1 4 2 3 6 4 Use Lagrange polynomials to find matrices E 1, E 2 R 3 3 so that A = E 1 +2E 2, E 1 + E 2 = I and E 1 E 2 = 0 (126) Let T be a linear operator on V which commutes with every projection operator on V What can you say about T? 9 Primary decomposition, cyclic subspaces and Jordan form (127) Let T be a linear operator on the finite-dimensional vector space V with characteristic polynomial c T (x) = k i=1 (x c i) d i and minimal polynomial m T (x) = k i=1 (x c i) r i Let W i = Null(T c i I) r i Show W i = {u V (T c i I) m u = 0 for some m depending on u} and dim W i = d i

(128) Let T be a rank one linear operator on a finite dimensional vector space V Show that either T is diagonalizable or T is nilpotent but not both (129) Show that two 3 3 nilpotent matrices over a field F are similar if and only if they have same minimal poynomials (130) Give an example of two 4 4 nilpotent matrices which have same minimal polynomials but which are not similar (131) Let T be the linear map induced on R 3 by the matrix diag(2, 2, 1) Show that T has no cyclic vector (132) Let T be a diagonalizable linear operator on an n-dimensional vector space V (a) If T has a cyclic vector, show that the characteristic polynomial has n distinct roots (b) If T has n distinct eigenvalues and {u 1, u 2,, u n } is a basis of eigenvectors then u = u 1 + u 2 + + u n is a cyclic vector for T (133) Let A be a complex 5 5 matrix with characteristic polynomial f(x) = (x 2) 3 (x+7) 2 and minimal polynomial p(x) = (x 2) 2 (x + 7) What is the Jordan form of A? (134) Let V be the complex vector space of polynomials of degree atmost 3 Let D : V V be the differentiation operator Find the Jordan form of the matrix of D in the standard basis B = {1, x, x 2, x 3 } of V (135) Let N be a k k nilpotent matrix whose degree of nilpotency is k Show that N t is similar to N Show that every complex n n matrix is similar to its transpose (136) Let N V = F n n be a nonzero nilpotent matrix with N n = 0 but N n 1 0 Show that there is no matrix A V such that A 2 = N (137) Let N be a 3 3 complex nilpotent matrix Prove that A = I + 1N 1N 2 satisfies 2 8 A 2 = I + N We say that A is a square root of I + N Let N be any n n complex nilpotent matrix Find a formula for square root of I + N (138) Use Jordan form to prove that every invertible n n complex matrix has a square root (139) Find the Jordan canonical form over C of the matrices: 2 1 1 1 ] 0 0 1 1 0 0 0 1 0 (140) Let A R n n such that A 2 + I = 0 Prove that n = 2k for some k N and A is similar over R to the matrix in block form: ] 0 I I 0 where I is the k k identity matrix 13

14 10 Spectral Theory and its applications Linear functionals and adjoints (141) Let V be a finite dimensional inner product space and T a linear operator on V Show that T (V ) = Null(T ) (142) Let V be an inner product space and v, w V be fixed vectors Define T (u) = u, v w Show that T has an adjoint and find T Now let V = C n Find the rank of T and the matrix of T in standard basis of C n (143) Let V be the vector space of real polynomials of degree atmost 3, with the inner product f(t), g(t) = 1 f(t)g(t)dt Let r R and let T (f(t)) = f(r) Show that 0 T is a linear functional and find g(t) V such that T (f(t)) = f(t), g(t) for all f(t) V (144) Let D be the differentiation operator on V as defined in the problem 143 Find D (145) Let V = C n n with inner product A, B = tr(ab ) Let P V be a fixed invertible matrix and define T (A) = P 1 AP for all A V Find the adjoint of T (146) Let T be a linear operator on a finite dimensional inner product space V Show that T is self-adjoint if and only if T u, u R for all u V Unitary operators (147) Let V be as in problem 145 For a fixed M V, Define T (A) = MA Show that T is unitary if and only if M is a unitary matrix (148) Let V = R 2 with standard inner product Let U be a unitary operator on V Let E be the standard basis of V Show that ] ] cos θ sin θ cos θ sin θ U] E = U θ = or sin θ cos θ sin θ cos θ Find the adjoint of U θ (149) Let V = R 2 with standard inner product Let W be the plane spanned by u = (1, 1, 1) t and v = (1, 1, 2) t Let U be the anticlockwise rotation through an angle θ about the line perpendicular to W when seen from a high point above the plane Find the matrix of U in the standard basis of R 2 (150) Let V be a finite dimensional inner product space and W a subspace of V Then V = W W Let u = v + w where u V, v W, and w W Define U(u) = v w Prove that U is self-adjoint and unitary Prove that every self-adjoint unitary operator on V arises this way from a subspace W of V Let W be the linear span of (1, 0, 1) t Find the matrix of U in the standard basis

(151) Let V be an inner product space A function T : V V is called a rigid motion if T (u) T (v) = u v for all u, v V (a) Show that unitary operators are rigid motions (b) Let w V be fixed Define T w, translation by w, by T w (u) = u + w for all u V Show that translations are rigid motions (c) Let V = R 2 Let T be a rigid motion of V with T (0) = 0 Show that T is linear and unitary (d) Show that every rigid motion of R 2 is translation followed by a rotation or a reflection and a rotation (152) For Normal operators A = 1 2 3 2 3 4 3 4 5 find an orthogonal matrix P such that P t AP is a diagonal matrix (153) Find an orthonormal basis of C 2 with standard inner product consisting of eigenvectors of the normal matrix A = 1 i i 1 (154) A linear operator T on an inner product space V is called positive if T is self-adjoint and T u, u is a positive real number Show that a normal linear operator T on a finite-dimensional inner product space V is self-adjoint, positive or unitary according as every eigenvalue of T is real, or positive or has absolute value 1 Find all positive and unitary operators on V (155) Show that a linear operator T defined on an inner product space V is normal if and only if T = T 1 + it 2 where T 1 and T 2 are self-adjoint operators on V such that T 1 T 2 = T 2 T 1 (156) Show that a real symmetric matrix has a real symmetric cube root (157) Let T be a normal operator on a finite dimensional inner product space Show that there is a complex polynomial f(x) such that T = f(t ) (158) Let A R n n be a symmetric matrix with A k = I for some k Show that A 2 = I (159) Show that if a normal linear operator T has spectral decomposition T = k i=1 a ie i then for any polynomial f(x) Cx], the spectral decomposition of f(t ) is given by f(t ) = k i=1 f(a i)e i (160) Using spectral theorem for symmetric matrices draw the conic section 9x 2 + 24xy + 16y 2 20x+15y = 0 and the quadric surface 7x 2 +7y 2 2z 2 +20yz 20zx 2xy = 36 ] 15