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Questions and Answers of Linear Algebra

Choose one from the following list of inner products on R2. Then find the adjoint ofwhen your inner product is used on both its domain and target space. (a) The Euclidean dot product (b) The weighted
Let M be a positive definite matrix. Show that A: Rn → Rn is self-adjoint with respect to the inner product (v, w) = vT Mw if and only if M A is a symmetric matrix.
Prove thatis self-adjoint with respect to the weighted inner product (v, w) = 2v1w1 + 3v2w2.
Consider the weighted inner product (v, w) = v1w1 + 1/2v2w2 + 1/3v3w3 on R3. (a) What are the conditions on the entries of a 3 × 3 matrix A in order that it be self-adjoint? (b) Write down an
Answer Exercise 7.5.12 for the inner product based onExercise 7.5.12 Consider the weighted inner product (v, w) = v1w1 + 1/2v2w2 + 1/3v3w3 on R3. (a) What are the conditions on the entries of a 3
True or false: A diagonal matrix is self-adjoint for any inner product on Rn.
Suppose L: U → U has an adjoint L*: U → U. (a) Show that L + L* is self-adjoint. (b) Show that L o L* is self-adjoint.
(a) Suppose K, M: U → U are self-adjoint linear functions on an inner product space U. Prove that (k[u], u) = (M[u], u) for all u ∈ U if and only if K = M. (b) Explain why this result is false if
Prove that if L: U → U is an invertible linear transformation on an inner product space U, then the following three statements are equivalent: (a) (L[u], L[v]) = (u, v) for all u, v ∈ U (b)
(a) Prove that the operation Ma[u(x)] = a(x)u(x) of multiplication by a fixed continuous function a(x) defines a self-adjoint linear operator on the function space C0[a, b] with respect to the L2
From the list in Exercise 7.5.1, choose different inner products on the domain and target space, and then determine the adjoint of the matrix A.Exercise 7.5.1Choose one from the following list of
A linear transformation S: U → U is called skew-adjoint if S* = -S. (a) Prove that a skew-symmetric matrix is skew-adjoint with respect to the standard dot product on Rn. (b) Under what conditions
Let L1: U → v1 and L2: U → V2 be linear maps between inner product spaces, with V1, V2 not necessarily the same. Let K1 = L1* ○ L1, K2 = L2* ○ L2. Show that the sum K = K1 + K2 can be written
Answer Exercise 7.5.24 for(a) The weighted norm ||(x, y)T|| = ˆš2x2 + 3y2;(b) The norm based on(c) The norm based on Exercise 7.5.24 Minimize ||(2x - y, x + y)T||2 - 6x over all x, y, where
LetMinimize p(x) = 1/2 ||L[x]||2 - (x, f) using (a) The Euclidean inner products and norms on both R2 and R3 (b) The Euclidean inner product on R2 and the weighted norm ||w|| = ˆšw21 + 2w22
How would you modify the statement of Theorem 7.62 if ker L ≠ {0}? Theorem 7.62 Suppose L: U → V is a linear map between inner product spaces with ker L = {0} and adjoint map L*: V → U. Let K =
Choose one from the following list of inner products on R3 for both the domain and target space, and find the adjoint of(a) The Euclidean dot product on R3 (b) The weighted inner product (v, w) =
From the list in Exercise 7.5.3, choose different inner products on the domain and target space, and then compute the adjoint of the matrix A.Exercise 7.5.3Choose one from the following list of inner
Choose an inner product on R2 from the list in Exercise 7.5.1, and an inner product on R3 from the list in Exercise 7.5.3, and then compute the adjoint ofExercise 7.5.1 Choose one from the following
Let p(2) be the space of quadratic polynomials equipped with the inner productFind the adjoint of the derivative operator D[p] = p€² acting on p(2).
Prove that, if it exists, the adjoint of a linear function is uniquely determined by (7.74).
Prove that (a) (L + M)* = L* + M* (b) (cL)* = cL* for c ∈ R (c) (L*)* = L (d) (L-1)* = (L*)-1
Show that the following linear transformations of R2 are self-adjoint with respect to the Euclidean dot product: (a) Rotation through the angle θ = π, (b) Reflection about the line y = x. (c) The
Solve the following initial value problems: (a) du/dt = 5u, u(0) = -3, (b) du/dt = 2u, u(1) = 3. (c) du/dt = -3m, m(-1) = 1.
Suppose that hunters are allowed to shoot a fixed number of the Northern Minnesota deer in Exercise 8.1.6 each year. (a) Explain why the population model takes the form du/dt = .27 u - b, where b is
(a) Prove that if ui (t) and u2 (t) are any two distinct solutions to du/dt = au with a > 0, then |u1(t) - u2(t)| → ∞ as t → ∞. (b) If a = .02 and u1(0) = .1, u2(0) = .05, how long do you
(a) Write down the exact solution to the initial value problem du/dt = 2/7u, u(0) = 1/3. (b) Suppose you make the approximation u(0) = .3333. At what point does your solution differ from the true
Let a be complex. Prove that u(t) = ceat is the (complex) solution to our scalar ordinary differential equation (8.1). Describe the asymptotic behavior of the solution as t → ∞, and the stability
Suppose a radioactive material has a half-life of 100 years. What is the decay rate γ? Starting with an initial sample of 100 grams, how much will be left after 10 years? 100 years? 1,000 years?
Carbon-14 has a half-life of 5730 years. Human skeletal fragments discovered in a cave are analyzed and found to have only 6.24% of the carbon-14 that living tissue would have. How old are the
Prove that if t* is the half-life of a radioactive material, then u(nt*) = 2-n u(0). Explain the meaning of this equation in your own words.
A bacteria colony grows according to the equation du/dt = 1.3 m. How long until the colony doubles? quadruples? If the initial population is 2, how long until the population reaches 2 million?
Consider the inhomogeneous differential equation du/dt = au + b, where a, b are constants. (a) Show that u* = - b/a is a constant equilibrium solution. (b) Solve the differential equation. (c)
Use the method of Exercise 8.1.7 to solve the following initial value problems: (a) du/dt = 2u - 1, u(0) = 1, (b) du/dt = 5u + 15, u(1) = -3. (c) du/dt = -3u + 6. u(2) = -1.
The radioactive waste from a nuclear reactor has a half-life of 1000 years. Waste is continually produced at the rate of 5 tons per year and stored in a dump site. (a) Set up an inhomogeneous
Find the eigenvalues and eigenvectors of the following matrices:
Let A be a given square matrix. (a) Explain in detail why any nonzero scalar multiple of an eigenvector of A is also an eigenvector. (b) Show that any nonzero linear combination of two eigenvectors
Let λ be a real eigenvalue of the real n × n matrix A, and v1, ..., vk a basis for the associated eigenspace Vλ Suppose w ∈ Cn is a complex eigenvector, so Aw = λw. Prove that w = c1 v1 + ... +
Define the shift map S: Cn → Cn by S(v1, v2, ..., vn-1, vn)T = (v2, v3, ..., vn, v1)T. (a) Prove that S is a linear map, and write down its matrix representation A. (b) Prove that A is an
(a) Compute the eigenvalues and corresponding eigenvectors of(b) Compute the trace of A and check that it equals the sum of the eigenvalues. (c) Find the determinant of A and check that it is equal
Verify the trace and determinant formulae (8.24- 25) for the matrices in Exercise 8.2.1.
(a) Find the explicit formula for the characteristic polynomial det(A - λI) = -λ3 + aλ2 - bλ + c of a general 3 × 3 matrix. Verify that a = tr A, c = det A. What is the formula for b? (b) Prove
Prove that the eigenvalues of an upper triangular (or lower triangular) matrix are its diagonal entries.
Let Ja be the n × n Jordan block matrix (8.22). Prove that its only eigenvalue is λ = a and the only eigenvectors are the nonzero scalar multiples of the standard basis vector e1.
(a) Find the eigenvalues of the rotation matrix(b) For what values of Î¸ are the eigenvalues real?
True or false: (a) If λ is an eigenvalue of both A and B, then it is an eigenvalue of the sum A + B. (b) If v is an eigenvector of both A and B. then it is an eigenvector of A + B.
Let A and B be n × n matrices. Prove that the matrix products A B and B A have the same eigenvalues.
(a) Prove that if λ ≠ 0 is a nonzero eigenvalue of A, then 1 / λ is an eigenvalue of A-1. (b) What happens if A has 0 as an eigenvalue?
(a) Prove that if det A > 1, then A has at least one eigenvalue with |λ| > 1. (b) If |det A| < 1, are all eigenvalues |λ| < 1? Prove or find a counter example.
Prove that A is a singular matrix if and only if 0 is an eigenvalue.
Prove that every nonzero vector 0 ≠ v ∈ Rn is an eigenvector of A if and only if A is a scalar multiple of the identity matrix.
How many unit eigenvectors correspond to a given eigenvalue of a matrix?
True or false: (a) Performing an elementary row operation of type #1 does not change the eigenvalues of a matrix. (b) Interchanging two rows of a matrix changes the sign of its eigenvalues. (c)
An elementary reflection matrix has the form Q = I - 2uu€², where u ˆˆ Rn is a unit vector.(a) Find the eigenvalues and eigenvectors for the elementary reflection matrices
Let A and B be similar matrices, so B = S-l A S for some nonsingular matrix S.(a) Prove that A and B have the same characteristic polynomial: pB(Î») = pA(Î»).(b) Explain why
Let A be a nonsingular n Ã— n matrix with characteristic polynomial pA(Î»).(a) Explain how to construct the characteristic polynomial pA-1(Î») of its inverse
A square matrix A is called nilpotent if Ak = O for some k ≥ 1. (a) Prove that the only eigenvalue of a nilpotent matrix is 0. (The converse is also true; see Exercise 8.6.18.) (b) Find examples
(a) Prove that every eigenvalue of a matrix A is also an eigenvalue of its transpose AT.(b) Do they have the same eigenvectors? Prove that if v is an eigenvector of A with eigenvalue Î» and w is an
(a) Prove that every real 3 × 3 matrix has at least one real eigenvalue. (b) Find a real 4 × 4 matrix with no real eigenvalues. (c) Can you find a real 5 × 5 matrix with no real eigenvalues?
(a) Show that if A is a matrix such that A4 = I. then the only possible eigenvalues of A are 1. -1, i and -i. (b) Give an example of a real matrix that has all four numbers as eigenvalues.
A projection matrix satisfies P2 = P. Find all eigenvalues and eigenvectors of P.
Let Q be an orthogonal matrix. (a) Prove that if λ is an eigenvalue, then so is 1/λ. (b) Prove that all its eigenvalues are complex numbers of modulus |λ| = 1. In particular, the only possible
(a) Prove that every 3 × 3 proper orthogonal matrix has +1 as an eigenvalue. (b) True or false: An improper 3 × 3 orthogonal matrix has -1 as an eigenvalue.
(a) Show that the linear transformation defined by a 3 Ã— 3 proper orthogonal matrix corresponds to rotating through an angle around a line through the origin in R3-the axis of the
Find all invariant subspaces, cf. Exercise 7.4.32, of a rotation in R3. Exercise 7.4.32 The subspace W of a vector space V is said to be an invariant subspace under the linear transformation L: V →
Suppose Q is an orthogonal matrix. (a) Prove that K = 2 I - Q - QT is a positive semi-definite matrix. (b) Under what conditions is K > 0?
Prove that every proper affine isometry F(x) = Q x + b of R3, where det Q = +1, is one of the following: (a) A translation x + b, (b) A rotation centered at some point of R3, or (c) A screw
Let Mn be the n Ã— n tridiagonal matrix whose diagonal entries are all equal to 0 and whose sub- and super-diagonal entries all equal 1.(a) Find the eigenvalues and eigenvectors of M2 and
Find a formula for the eigenvalues of the tricirculant n × n matrix Zn that has 1's on the sub- and super-diagonals as well as its (l, n) and (n, 1) entries, while all other entries are 0.
(a) Write out the characteristic equation for the matrix(b) Show that, given any 3 numbers a, b, and c, there is a 3 x 3 matrix with characteristic equation - Î»3 + a Î»2 + b
Let A be an n Ã— n matrix with eigenvalues Î»1,..., Î»k. and B an m Ã— m matrix with eigenvalues Î¼1, ..., Î¼1. Show that the
Deflation: Suppose A has eigenvalue Î» and corresponding eigenvector v.(a) Let b be any vector. Prove that the matrix B = A - vbT also has v as an eigenvector, now with eigenvalue
Find the eigenvalues and eigenvectors of the cross product matrix
Find all eigenvalues and eigenvectors of the following complex matrices:(a)(b) (c) (d)
Find all eigenvalues and eigenvectors of (a) The n x n zero matrix O; (b) The n x n identity matrix I.
Find the eigenvalues and eigenvectors of an n x n matrix with every entry equal to 1.
Which of the following are complete eigenvalues for the indicated matrix? What is the dimension of the associated eigenspace?
(a) Prove that if λ is an eigenvalue of A, then λn is an eigenvalue of An. (b) State and prove a converse if A is complete. (The completeness hypothesis is not essential, but this is harder,
Show that each eigenspace of an n × n matrix A is an invariant subspace, as defined in Exercise 7.4.32. Exercise 7.4.32 The subspace W of a vector space V is said to be an invariant subspace under
(a) Prove that if x ± iy is a complex conjugate pair of eigenvectors of a real matrix A corresponding to complex conjugate eigenvalues μ ± iv with v ≠ 0, then x and y are linearly independent
Diagonalize the following matrices:(a)(b) (c) (d) (e) (f) (g) (h) (i) (j)
Diagonalize the Fibonacci matrix
Diagonalize the matrixof rotation through 90°. How would you interpret the result?
Diagonalize the rotation matrices(a)(b)
Which of the following matrices have real diagonal forms?(a)(b) (c) (d) (e) (f)
Find the eigenvalues and a basis for the each of the eigenspaces of the following matrices. Which are complete?
Diagonalize the following complex matrices:(a)(b) (c) (d)
Write down a real matrix that has(a) Eigenvalues -1, 3 and corresponding eigenvectors(b) Eigenvalues 0, 2, -2 and associated eigenvecors (c) An eigenvalue of 3 and corresponding eigenvectors (d) An
A matrix A has eigenvalues -1 and 2 and associated eigenvectorsWrite down the matrix form of the linear transformation L[u] = A u in terms of (a) The standard basis e1, e2; (b) The basis consisting
Prove that two complete matrices A. B have the same eigenvalues (with multiplicities) if and only if they are similar, i.e., B = S-1 AS for some nonsingular matrix S.
Let B be obtained from A by permuting both its rows and columns using the same permutation π, so bij = aπ(i),π(j). Prove that A and B have the same eigenvalues. How are their eigenvectors related?
True or false: If A is a complete upper triangular matrix, then it has an upper triangular eigenvector matrix S.
Suppose the n × n matrix A is diagonalizable. How many different diagonal forms does it have?
Characterize all complete matrices that are their own inverses: A-1 = A. Write down a nondiagonal example.
Two n × n matrices A, B are said to be simultaneously diagonalizable if there is a nonsingular matrix S such that both S-1 A S and S-1 B S are diagonal matrices. (a) Show that simultaneously
Which of the following matrices admit eigenvector bases of Rn? For those that do. Exhibit such a basis. If not, what is the dimension of the subspace of Rn spanned by the eigenvectors?
(a) Give an example of a 3 × 3 matrix that only has 1 as an eigenvalue, and has only one linearly independent eigenvector. (b) Give an example that has two linearly independent eigenvectors.
(a) Prove that if A is complete, so is A2. (b) Give an example of an incomplete matrix A such that A2 is complete.
Suppose v1, ..., vn forms an eigenvector basis for the complete matrix A, with λ1, ..., λn the corresponding eigenvalues. Prove that every eigenvalue of A is one of the λ1, ..., λn.
Find the eigenvalues and an orthonormal eigenvector basis for the following symmetric matrices:(a)(b) (c) (d) (e)

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