Linear Algebra. Workbook

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1 Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011

2 Checklist Name: A B C D E F F G H I J xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx

3 Problem A1. Prove that a vector space cannot be the union of two proper subspaces.

4 Problem A2. Show that two subsets A and B of a vector space V generate the same subspace if and only if each vector in A is a linear combination of vectors in B and vice versa.

5 Problem A3. Show that {1, x, x 2,..., x n,... } is a basis of F[x].

6 Problem A4. Let H be the quaternion algebra. Let x = a+bi+cj +dk H. Define its conjugate to be the quaternion x = a bi cj dk. (i) Show that xx = xx = (a 2 + b 2 + c 2 + d 2 ). (ii) Show that H is a division algebra, i.e., xy = 0 = x = 0 or y = 0. (ii) Let q be a nonzero quaternion. Show that {q, qi, qj, qk} is a basis of H.

7 Problem A5. Complete the following table to verify the associativity of the quaternion algebra H. a b c bc a(bc) ab (ab)c i i j k ik = j 1 j i j i j i i i j j j i j j j i i j k i k j Why is this enough?

8 Problem A6. (a) If X is a subspace of V, then dimx dimv. (b) If dimv = n, the only n-dimensional subspace of V is V itself.

9 Problem A7. Let A = {a 1,..., a r } be a set of vectors such that each subset of k vectors is linearly dependent. Prove that Span(A) is at most (k 1)-dimensional.

10 Problem A8. Let V be a vector space over F, and A a subspace of V. Two vectors x, y V have two linearly independent linear combinations which are in A. Show that x and y are both in A.

11 Problem A9. Let F be an infinite field. Prove that a vector space over F cannot be a finite union of proper subspaces.

12 Problem A10. Let a, b, c R, not all zero, and Name: X = {(x, y, z) R 3 : ax + by + cz = 0}. (a) Show that X is a subspace of R 3. (b) Find a subspace Y such that R 3 = X Y, and deduce that X is 2-dimensional.

13 Problem B1. What are the inverses of the elementary matrices E i,j, E i (λ), and E i,j (λ)?

14 Problem B2. Let M be an n n matrix over F that can be brought to the identity matrix by a sequence of elementary row operations. Show that M is invertible, and find its inverse.

15 Problem B3. Let A and B be n n matrices such that AB = I n. Show that BA = I n.

16 Problem B4. Find the inverse of the n n matrix A = (a i,j ) where { 1 if i j, a i,j = 0, if i > j.

17 Problem B5. Consider a system of linear equations Ax = b. Suppose the entries of the matrices A and b are all rational numbers, and the system has a nontrivial solution in C. Show that it has a nontrivial solution in Q.

18 Problem B6. (corrected) Assume that the system of equations cy+bz = a, cx +az = b, bx+ay = c has a unique solution for x, y, z. Prove that abc 0 and find the solution.

19 Problem B7. (corrected) Given p, q R, not both zero, consider the system of equations py + qz = a, qx +pz = b, px + qy = c in x, y, z. (a) Show that the system has a unique solution if and only if p + q 0. In this case, find the solution. (b) Suppose p +q = 0. Show that the system has solutions if and only if a + b + c = 0. (c) Solve the system of equations in each of the following cases: (i) p = q = 1; (ii) p = 1, q = 1.

20 Problem B8. (a) Find the determinant of the matrix a a a + 1 (b) Consider the system of equations (a + 1)x + y + z = p, x +(a + 1)y + z = q, x + y + (a + 1)z = r in x, y, z. Solve the system in the following cases: (i) a 0, 3; (ii) a = 0; (iii) a = 3.

21 Problem B9. (corrected) Let a 1,..., a n be given numbers. Compute the determinant of the n n matrix A = (a ij ), where a ij = a i 1 j.

22 Problem C1. Let f : X Y be a linear transformation. If x 1,..., x n are linearly dependent, show that f(x 1 ),..., f(x n ) are linearly dependent.

23 Problem C2. Show that two finite dimensional vector spaces over F are isomorphic if and only if they have equal dimensions.

24 Problem C3. (corrected) Let q = a + bi + cj + dk be a nonzero quaternion. (a) Prove that the left multiplication L q : H H given by L q (x) = qx is an isomorphism. (b) Find the matrix of L q relative to the basis 1, i, j, k H.

25 Problem C4. Let s : l 2 l 2 be the map Name: s(x 0, x 1, x 2,...) = (0, x 0,...) (a) Is this a linear map? (b) Is s a monomorphism? (c) Is s an epimorphism?

26 Problem C5. (Previously mislabeled C4) Let X be a finite dimensional vector space over F, and f : X X a linear transformation. If the rank of f 2 = f f is equal to the rank of f, prove that kerf Image f = {0}.

27 Problem C6. Let f, g : X Name: Y be linear transformations. Prove that rank(f + g) rank(f) + rank(g).

28 Problem C7. Let V be a finite dimensional vector space over F. If f, g : V V are linear transformations such that g f = ι V, prove that f g = ι V.

29 Name: Problem C8. Given linear transformations g : Y Z and h : X Z, prove that there exists a linear transformation f : X Y such that h = g f if and only if Image h Image g. X h Z f g Y

30 Problem C9. Let X and Y be vector subspaces of V such that V = X Y. Prove that V/X and Y are isomorphic.

31 Problem C10. Let X = X 1 X 2, and V = V 1 V 2 with V 1 X 1 and V 2 X 2. Prove that X/V (X 1 /V 1 ) (X 2 /V 2 ).

32 Problem D1. Let f : X Y be a linear transformation. Show that f : Hom F (Y, Z) Hom F (X, Z) defined by is a linear transformation. f (g) = g f

33 Problem D2. (a) Let X and Y be vector spaces over F. Describe the vector space structure (over F ) of X Y. (b) Given f : V X and g : V Y, show that there is a unique linear map h : V X Y such that π 1 h = f and π 2 h = g.

34 Problem D3. Let B = {e 1, e 2, e 3, e 4 } be a basis of a 4-dimensional space V over F, and B = {e 1, e 2, e 3, e 4 } its dual basis. Find the dual basis of C = {e 1, e 1 + e 2, e 1 + e 2 + e 3, e 1 + e 2 + e 3 + e 4 } in terms of e 1,..., e 4.

35 Problem D4. Let n be an odd number. Decide if the permutation ( ) 1 2 k n 2 4 2k 2n is an even or odd permutation. Here in the second row, the values are taken modulo n. (Thus, 2n is replaced by n).

36 Problem D5. Let M be the space of 2 2 matrices over F, and ( ) a b A =. c d Compute the determinant of the linear transformation Φ : M M defined by Φ(X) = AX.

37 Problem D6. Prove that the isomorphism Φ X : X Name: X defined by Φ X (x)(ϕ) = ϕ(x) for x X, ϕ X is natural in the sense that for every linear transformation f : X Y, the diagram X f Y Φ X Φ Y is commutative. X Y X f Y

38 Problem D7. Here is Puzzle 128 of Dudeney s famous 536 Curious Problems and Puzzles. Take nine counters numbered 1 to 9, and place them in a row in the natural order. It is required in as few exchanges of pairs as possible to convert this into a square number. As an example in six moves we give the following: ( ), which give the number , which is the square of But it can be done in much fewer moves. The square of can be found in four moves, as in the first example below. The squares of 25572, 26733, and can also be obtained in four moves (25) (37) (38) (34) However, there is one, which is the square of a number of the form aabbc, which can be made in three moves. Can you find this number?

39 Problem E1. 1 Let A be a 3 3 matrix. Prove that the characteristic polynomial of A is det(a λi) = λ 3 +(a 11 +a 22 +a 33 )λ 2 (A 11 +A 22 +A 33 )λ+deta. 1 Typo corrected: The constant term was previously set as deta.

40 Problem E2. Let λ be an eigenvalue of M. Prove that λ k is an eigenvalue of M k for every integer k 1.

41 Problem E3. Let M B (f) = Find the eigenvalues and eigenvectors of f and show that it is diagonalizable.

42 Problem E4. Let M B (f) = Find the eigenvalues and eigenvectors of f and show that it is not diagonalizable.

43 Problem E5. Diagonalize the linear operator on R 3 : M B (f) =

44 Problem E6. (a) Let M be the n n matrix whose entries are all 1 s. Find the eigenvectors and eigenvalues of M. (b) Let M be the n n matrix whose diagonal entries are all zero, and off diagonal entries all 1 s. Compute det M.

45 Problem E7. Find the eigenvalues and eigenvectors of the right shift operator R on l 2 : R(x 0, x 1, x 2,...) = (0, x 0, x 1,...).

46 Problem E8. Let V be a finite dimensional vector over F, and f, g : V linear operators on V. Prove or disprove (a) Every eigenvector of g f is an eigenvector of f g. (b) Every eigenvalue of g f is an eigenvalue of f g. V

47 Problem E9. Complete the details for the proof of dim C V C = dim R V.

48 Problem E10. Let V be a complex vector space, with conjugation χ : V V. Prove that a subspace W V is the complexification of a real vector space S if and only if W is closed under χ.

49 Problem F1. Prove that a positive definite inner product is nondegenerate: if x V is such that x, y = 0 for every y V, then x = 0.

50 Problem F2. Prove the triangle inequality in a positive definite inner product space: For x, y V, x + y x + y. Equality holds if and only if x and y are linearly dependent.

51 Problem F3. (Typos corrected) Verify the polarization identity in a real inner product space: for u, v V, u, v = 1 ( u + v 2 u v 2). 4

52 Problem F4. Let (a, b, c) be a unit vector of R 3. (a) Find the orthogonal projection of the vector (u, v, w) R 3 on the plane ax + by + cz = 0. (b) Find the matrix, relative to the standard basis e 1, e 2, e 3, of the orthogonal projection π : R 3 R 3 onto the plane ax + by + cz = 0.

53 Problem F5. Prove that if the columns of an n n matrix with entries in C are mutually orthogonal unit vectors, then so are the rows.

54 Problem F6. Let a, b R. Prove that there are two orthogonal unit vectors u, v R 3 of the form u = (u 1, u 2, a) and v = (v 1, v 2, b) if and only if a 2 + b 2 1.

55 Problem F7. Let u = (a 1, b 1, c 1 ) and v = (a 2, b 2, c 2 ) be nonzero vectors of R 3. Find a basis of the orthogonal complement of the span of u and v.

56 Problem F8. (corrected) In the space of continuous functions over [ 1, 1] consider the inner product f, g := 1 1 f(x)g(x)dx. For each n = 1, 2, 3, 4, find a monic polynomial p n (x) of degree n and with leading coefficient 1, orthogonal to all polynomials of lower degrees.

57 Problem F9. (new) Prove that for every n n matrix A, I n +A t A is positive definite, and is nonsingular.

58 Problem F10. (new) Let V be a vector space with a positive definite inner product, and f : V V a mapping satisfying f(x), f(y) = x, y for all x, y V. Prove that f is a linear transformation.

59 Problem F11. (new) Find the angles between the two 2-planes: A : y = Ax and B : y = Bx, where A = ( ) and B = ( )

60 Problem F12. Find the equation of the 2-plane in R 4 containing the vector (1, 0, 0, 0) and isoclinic with the 2-plane ( ) 1 0 y = x. 0 0

61 Problem F13. Prove that L a = L t a for a quaternion a.

62 Problem G1. Let π : V V be a linear transformation satisfying π 2 = π. Prove that π is the projection of V onto Image π along kerπ.

63 Problem G2. A linear transformation f : V V is called (i) an idempotent if f 2 = f, (ii) an involution if f 2 = ι V. Prove that f is an idempotent if and only if g := 2f ι is an involution.

64 Problem G3. Let f : V V be a linear transformation. Prove that the induced map f : Hom(V, V ) Hom(V, V ) defined by f (g) = f g is a projection if and only if f is a projection.

65 Problem G4. Let f : V V be a given linear transformation. A subspace X V is invariant under f if f(x) X. (i) Let X be a subspace invariant under f. Prove that for every projection π = π X,Y, π f π = f π. (ii) Suppose the relation in (i) holds for some direct sum decomposition V = X Y. Prove that X is invariant under f.

66 Problem G5. Prove that the Riesz map R : V is an isomorphism. Name: V defined by R(f), y = f(y) for every y V,

67 Problem G6. Let f : X Y be a linear transformation between inner product spaces. Show that f = f.

68 Problem G7. Without making use of the Theorem in 7B, prove that if f : V V is a self adjoint linear operator, there is an orthonormal basis of V consisting of eigenvectors of f.

69 Problem G8. Let e 1 = (1, 0) C 2 regarded as a Hermitian space. Find all unit vectors in C 2 orthogonal to e 1.

70 Problem G9. Prove the hyperplane reflection formula τ v (x) = x + 2 x, v, v.

71 Problem G10. Given a unit vector v = (a, b, c) R 3, find the matrix, relative to the standard basis of R 3, of the rotation about v by 180 degrees.

72 An n n matrix over C is (i) Hermitian if A = A, (ii) skew-hermitian if A = A, (iii) unitary if A A = I = AA.

73 Problem( H1. ) 1 1 Let A =. 1 1 (a) Show that A is normal. (b) Is A Hermitian? skew-hermitian? or unitary?

74 Problem H2. Let U be a unitary matrix. (a) Let A be a normal matrix. U AU is normal. (b) Let A be a Hermitian matrix. Show that U AU is Hermitian. (c) Let A be a unitary matrix. Show that AU is unitary.

75 Problem H3. Let A be a diagonal matrix. (a) Show that A is Hermitian if and only if its entries are real. (b) Show that A is skew-hermitian if and only if its diagonal entries are purely imaginary. (c) Show that A is unitary if and only if its diagonal entries are unit complex numbers.

76 Problem H4. Prove that the trace of a matrix is a similarity invariant: if A, P M n (F) and P nonsingular, then Tr(P 1 AP) = Tr(A).

77 Problem H5. Let A = (a i,j ) and B = (b i,j ) be complex n n matrices. Prove that if A and B are unitarily equivalent, then i,j a i,j 2 = i,j b i,j 2.

78 Problem H6. Let A = (a i,j ) be a complex matrix with eigenvalues λ 1,..., λ n. Prove that if A is normal, then i,j a i,j 2 = i λ i 2.

79 Problem H7. Show that the matrices A = and B = are similar and satisfy i,j a i,j 2 = i,j b i,j 2, but are not unitarily equivalent.

80 Problem H8. Let A and B be similar 2 2 matrices satisfying i,j a i,j 2 = i,j b i,j 2. Prove that A and B are unitarily equivalent.

81 Problem H9. Let A be a nonsingular matrix. Prove that every matrix which commutes with A also commutes with A 1.

82 Problem H10. Let A be an n n matrix for which A k = 0 for some k > n. Prove that A n = 0.

83 Problem I1. A linear transformation f : R Name: R 3 has matrix relative to the standard basis e 1, e 2, e 3. Show that f is an isometry.

84 Problem I2. Find the invariant 1-planes and invariant 2-planes of the isometry f in Problem I1 with matrix relative to the standard basis e 1, e 2, e 3.

85 Problem I3. Let f be an isometry of R 4 with matrix cosθ sin θ sin θ cosθ cosθ sin θ, sin θ cosθ with 0 < θ < 2π relative to the standard basis e 1, e 2, e 3, e 4. Show that f rotates every vector of R 4 by an angle θ.

86 Problem I4. Let r and s be orthogonal pure unit quaternions. (a) Show that r, s, rs is an orthonormal basis of the 3-plane i, j, k. (b) Show that the solution space of the equation ru = ur is the span of 1 and r. (c) Show that the solution space of the equation ru = ur is the orthogonal complement of Span(1,r).

87 Problem I5. Let a H be a fixed unit quaternion. Consider the isometry f a : H H defined by f(u) = aua. (a) Prove that f leaves the 3-plane W := Span(i, j, k) invariant. (b) Prove that f induces an orientation preserving isometry on W.

88 Problem I6. Use the Cayley-Hamilton theorem to find the inverse of a nonsingular 2 2 matrix ( ) a b. c d

89 Problem I7. Compute A 10 for the matrix A =

90 Problem I8. Find a real matrix B such that B 2 =

91 Problem I9. Prove or disprove: for any 2 2 matrix A over C, there is a 2 2 matrix B such that A = B 2.

92 Problem I10. For x R, let Name: x A x := 1 x x x (a) Prove that det(a x ) = (x 1) 3 (x + 3). (b) Prove that if x 1, 3, then A 1 1 x = (x 1)(x + 3) A x 2.

93 Problem J1. Find the Jordan canonical form of the all-one matrix of order n, i.e., A = (a i,j ) with a i,j = 1 for all i, j = 1,..., n), given that its eigenvalues are n (multiplicity 1) and 0 (multiplicity n 1).

94 Problem J2. A 6 6 matrix A has eigenvalue λ of multiplicity 6, and the eigenspace V λ has dimension 3. What are the possible Jordon canonical forms?

95 Problem J3. Determine all possible Jordan canonical forms for a matrix with characteristic polynomial (λ 3) 6 and minimal polynomial (λ 3) 2.

96 Problem J4. What are the possible Jordan canonical forms for a matrix of order 6 whose characteristic polynomial is (λ 2) 4 (λ + 3) 2?

97 Problem J5. Find the canonical form of the matrix

98 Problem J6. Find a nonsingular matrix P such that, for the matrix A = , P 1 AP is a Jordan canonical form.

99 Problem J7. Let T be an upper triangular matrix over R which commutes with its transpose. Show that T is a diagonal matrix.

100 Problem J8. ( λ 1 Show that if A = 0λ A n = for every positive integer n. ), then Name: ( ) λ n nλ n 1 0 λ n

101 Problem J9. Let A be a 2 2 complex matrix. Show that the series I + A + A A n + converges if and only if λ < 1 for each eigenvalue λ of A.

102 Problem J10. Let A and B be real 2 2 matrices such that A 2 = B 2 = I, and AB + BA = 0. Prove that there exists a real nonsingular matrix P such that P 1 AP = ( ) and P 1 BP = ( )

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