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Differential Equations and Linear Algebra 2nd edition Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West - Solutions

Use the definition of projection, as stated here, for problems a to b A linear transformation T: V → W, where W is a sub-space of V, is called a projection provided that T, when restricted to W, reduces to the identity mapping; that is For all vectors w in the subspace W. a. Verify that the
A mapping T: R2 †’ R2 is given byWhereShow that T rotates every vector v Îµ R2 counterclockwise about the origin through angle 0. Express v using polar coordinates, And use the identities for cos(Î¸ + Î±) and sin(Î¸ + Î±).
Integral Transforms If K(s, t) is a continuous function of s and t on the square 0 ‰¤ s ‰¤ 1, 0 ‰¤ t ‰¤ 1, and i(r)is any continuous function of t for 0 ‰¤ t ‰¤ 1. we can define the function F given byShow that the mapping T(f
Computer Lab: Matrix Machine Use the matrix machine from IDE (Lab 15) to analyze each transformation from R2 to R2 in Problems 1 to 2, and answer the following questions.(a) Which vectors are not moved by the transformation?(b) Which nonzero vectors do not have their direction changed?(c) Which
Find the kernel for the linear transformation in Problems a to b Describe the kernel. a. T: R2 → R2, T(x, y) = (-x, y) b. T: R3 → R2, T(x, y, z) = (2x + 3t - z, -x + 4y + 6z)
Calculus Kernels The transformations in Problems 1 to 3 should be familiar from calculus. Identify each transformation and give its kernel. 1. T: P2 → P2. T(at2 + bt + c) = 2at + b 2. T: P2→ P2. T(at2 bt + c) = 2a 3. T: P2 → P2. T (at2 + bt + c) = 0
For Problems a to b, suppose that T: V → W is a linear transformation from vector space V to vector space W. Also suppose that w is a solution of T(u) = b1, and that u1 is a solution of T(u) = b2. Then u1 + u2 is a solution of T(u) = b1 + b2; this is called the Superposition Principle. as first
Dissecting Transformations In each of Problems 1 to 2, a transformationgiven by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is infective or surjective. 1. 2.
Let T: R2 †’ Rm be a linear transformation, and letBe a linearly dependent set in Rn. Prove that the set Is linearly dependent in Rm
Let T: Rn †’ Rm be linear transformation, and letBe a linearly dependent set in Rn. Give a counterexample to show that Need not be linearly independent in Rm.
Let T: Rn †’ Rm be an injective linear transformation, and letbe a linearly independent set in Rn. Prove that must be a linearly independent set in JR"' .
Prove that if a linear transformation T maps two linearly independent vectors onto a linearly dependents set, the n the equationhas a nontrivial solution.
Consider the transformation T: P2 †’ R2 defined byFor example, if p(t) = t2 - 6t + 4, then a Prove that T is a linear transformation. b Find a basis for the kernel of T. c Find a basis for the image of T.
Find the kernel and image of each linear transformation in Problems a to c. a. T: M22 → M22, T(A) = AT b. T: P3 → P3, T(p) = p'
Give examples of matrices A in M33 such thatHas the properties described in Problems a to b a. The Im(T) is the plane 2x - 3y + z = 0. b. The Im(T) is the line spanned by
Answer problems a to c true or false, and give a brief explanation or counterexample. a. If A is a square matrix, then Ker(A2) = Ker{A). True or false? b. If A is a square matrix, then Im(A2) = Im(A). True or false? c. If A is a square matrix, then Ker(A) = Ker(RREF). True or false?
A transformation T: R4 †’ R2 is de-fined with matrix multiplication to be T(v) = Ai It is known that the RREF of A isDetermine dim(Ker( T)) and d.im(lm( T)). Is T one-to one? Is it onto R2? Find bases for the kernel and image.
Consider the transformation T: R2 †’ R4 defined byThe RREF of B is Determine dim(Ker(T)) and dim(lm(T)). Is T one-to one? Is it onto R4?
For the transformation T: R3 †’ R4 defined byWhere A has RREF Determine dim(Ker(T)) and dim(lm(T)). Is T one-to one? ls it onto R4?
Consider transformation T: R3 †’ R3 defined byWhere C has RREF Determine dim(Kcr(T)) and dim(lm(T)) of transformation T, and decide whether ii is injective and/or surjective.
Express the general solution for each system in problems a to c as the sum of a particular solution and the solution of the corresponding homogeneous systems. a. x + y = 1 b. = 3x - y + z = -4 c. x + 2y = 2
In each of Problems 1 to 3, express the general solution of the nonhomo-geneous DE as the sum of a particular solution and the general solution of the corresponding homogenenous equation. The homogeneous equations are linear or separable; particular solutions (mostly constant) may be found by
For each equation in Problems a to b, express the general solution of the nonhomogeneous DE a the sum of a particular solution (each is a polynomial in t) and the general solution of the corresponding homogeneous DE. a. y" + y' - 2y = 21 - 3 b. y" - 2y' + 2y = 4t - 6
For each matrix in Problems a to b compute its eigenvalues and eigevector(s), and sketch the eigenspaces when the eigenvectors are real.a.b.
Eigenvedor Shortcut For a 2 Ã— 2 matrixWith eigenvalue Î», show that if ‰ 0, then the corresponding eigenvector is
The eigenvector shortcut of Problem 17 may fail when h = 0, forcing a return to the definition Av = Î»v to find the eigenvector(s). For each eigen-value in the following matrices, find the) eigenvector (s). Discuss how/why the shortcut fails and why the definition succeeds.a.b. c.
For each matrix in Problems a to b compute its eigenvalues, eigenvectors and the dimension of each eigenspace.a.b.
Show that the set of eigenvectors belonging to a particular eigenvalue of an n x n matrix, together with the zero vector, is a subsiace of 1121.
Extend the proof of the Distinct Eigenvalue Theorem for a 3 × 3 matrix A as fol-lows: Show that if A has 3 distinct eigenvalues λ1 λ2. λ1, then the corresponding eigenvectors v1, v2, v3 are linearly independent. Use the fact that an eigenvector vi, can-not be zero, and follow the steps shown in
Show that an invertible matrix cannot have a zero eigenvalue. In fact, you will have proved a characteristic of invertible matrices.
Suppose that λ is an eigenvalue of an invertible matrix A. Show that 1/λ is an eigenvalue of A-1.
(a) Use the definition for similar matrices (i.e., B ~ A if and only if B = p-1 AP for some invertible matrix P) to show that similar matrices have the same characteristic polynomials and eigenvalues. (b) Show, using 2 × 2 matrices as examples, that the eigenvectors may be different for similar
What are the eigenvalues and eigenvectors of the following?What about I3? What about In?
If a matrix A has an inverse A-1, use equation (2) to show that A-1 has the same eigenvectors as A. Determine a relationship between the eigenvalues of A and A-1. Illustrate with a suitable example.
The eigenvalues of an upper triangular matrix and those of a lower triangular matrix appear on the main diagonal. Verify this fact for the matrices in Problems a to c.a.b. c.
Use properties of determinants (Sec. 3.4) to demonstrate why the diagonal eigenvalue property holds in general for triangular matrices.
For Problems a to b, let A be a square matrix and determine the following facts about its transpose. a. Show that A is invertible if and only if AT is invertible. b. Show that A and AT have the same eigenvalues.
Let A be a symmetric matrix (that is, A = AT) with distinct eigenvalues Î»1 and Î»2. For such a matrix, if v1 and v2 are eigenvectors belonging to the distinct eigenvalues Î»1 and Î»2, respectively, then v1 and v2 are orthogonal(a) Illustrate this
Find the eigenvalues, if any, and the corresponding eigenspaces for the linear transformation T: P1 → P2 defined by T(ax2 + bx + c) = bx + c.
In a quadratic equation with leading coefficient I, the negative of the coefficient of the linear term is the sum of the roots, and the constant term is the product of the roots.(a) Prove these properties by expanding the factored quadratic(x - Î»1)(x - Î»2) = 0.(b) Compare
For problem 1 to 3 find all the 2 Ã— 2 matrices with the desired properties.1.is an eigenvector. 2. is an eigenvector. 3. And Are eigenvectors,
For r Problems a to b find the eigenvalues, if any, add corresponding eigenvectors for the transformations in Table 5.1.1.(a) Reflections about the x-axis. (b) Reflections about the y-axis.
If Î»2 + bÎ» + c = 0 is the characteristic equation of the 2 Ã— 2 matrix A for example, then A2 + bA + cl =0. Verify this for each matrix in Problems a and b.a.b.
For an invertible 3 Ã— 3 ma-trix A, we can write, using the Cayley-Hamilton Theorem, A3 + bA2 + cA + dl = 0. where b, c, and d are coefficients of the characteristic equation of A. If we multiply through on the left by A-1, we get A2 + bA + cl + dA-1 = 0, which can be solved for A-1.
Develop a Cayley-Hamilton formula for the inverse of a 2 Ã— 2 matrix, and apply it to compute the inverses of the following(a)(b)
Express the eigenvalues of a 2 × 2 matrix in terms its trace and its determinant.
Generalize the results of Problem 70 to the characteristic equation and eigenvalues of a 3 Ã— 3 matrix. Then illustrate these results for the matrix
Using the method of Sec. 4.7, convert each differential equation in Problems 1-2 to a system o f first-order equationsm Then verify that the charac-teristic roots of the DE are the same as the eigenvalues of the matrix of the converted linear system. (1) y" - y' - 2y = 0 (2) y" - 2y' + 5y = 0
For what values of the nonnegative constant a in the equation y" + λy = 0 do thereexist nonzero solutions satisfying the boundary condi-tions in Problems a-b? The values of a are called eigenvaltions and the corresponding solutions are called eigenfunctions. (a) y(0) = 0, y(π) = 0 (b) y'(0) =
For Problems 1-2, the standard basis isLet be a new basis for R3. (1) Convert the vectors from the standard basis to basis B. (2) Vectors are expressed in basis B. Find their standard basis representations.
For Problems a-b, we take S = {x2, x, 1} as the standard basis in P2 and introduce a new basis N = {2x2 - x, x2, x2 + 1}(a) Express in terms of the new basis N the standard basis polynomials(b) Vector representations of three polynomials relative to the basis N are Calculate standard
Polynomial Coordinates II For Problems a to b. We take S = (x3, x2. x, I) as the standard basis in P3 and introduce a new basis Q = (x3, x3 + x, x2. x2 + I).(a) Find the coordinate vectors of these standard basis polynomials in terms of the new basis Q:(b) Vector representations of three
Using the standard basis S = {r4, t3, t3, t2, t, l} for P4, deter-mine a matrix representing the transformation from P. to P4 given in each of Problems. Then apply your matrix to the following polynomials: (a) g(t) = t4 - t3 + t2 - t + 1 (b) q(t) = t4 + 2t2 + 4 (c) r(t) = -4t4 + 3t3 (d) w(t) = t4 -
In Problems a to c, letbe the standared basis, and be a new basis for R2. (a) Calculate the coordinate-change matrices M8 to go from B to S and M-1B as in Example 2. (b) Convert the vectors And from the standard basis to the basis B. (c) Vectors And
In Problems 1 to 3. determine whether each matrix A is diagonalizable. If it is, determine a matrix P that diagonalizes it and compute P-1 AP. You can obtain P-1 AP directly from careful construction of a diagonal matrix with eigenvalues along the diagonal in the proper order.(1)(2) (3)
Suppose that A is a diagonalizable matrix that has been written in the form A = PDP-1, where D is diagonal.(a) Show that for positive integer k, Ak = PDkP-1.(b) Use the result of part (a) to compute A50° for(c) Show that if D is diagonal, then Dk is diagonal. (d) Is the equation in part (a)
In Problems 1-2, letbe the standard basis, and be a new basis for R2. (1) Convert the vectors And from the standard basis to the basis B. (2) Vectors And are expressed in basis B. Find their standard basis representations
Let A be an n Ã— n matrix such that its characteristic polynomial isWhere the Î» are distinct (a) Explain why (b) If the Î»i are not distinct but A is diagonalizable, would the same property hold? Explain.
In Problems a-c, construct the required examples. (a) Construct a 2 × 2 matrix that is invertible but not diagonalizable. (b) Construct a 2 × 2 matrix that is diagonalizable but not invertible. (c) Construct a 2 × 2 matrix that is neither invertible nor diagonalizable
Use appropriate computer software to diagonalize (if possible) the following matrices.(a)(b) (d)
We have defined matrix B to be similar to matrix A (denoted by B ~ A) if there is an invertible matrix P such that B = P-1AP. Prove the following: (a) Similar matrices have the same determinant and the same trace (b) Show by example (using 2 × 2 matrices) that similar matrices can have different
An arthogonal matrix P is a square matrix whose transpose equals its inverse:(a) Show that this is equivalent to the condition PPT = I. (b) Use part (a) to show that the column vectors of an orthogonal matrix are orthogonal vectors.
A matrix A is orthogonally diagonalizable if there is an orthogonal matrix P that diagonalizes it. Show that the matrixis orthogonally diagonalizable. Synunetric matrices have orthogonal eigenvectors.
For Problems a-b, the standard basis isLet be a new basis for R3. (a) Convert the vectors from the standard basis to the basis B. (b) Vectors are expressed in basis B. Find their standard basis representations.
For Problems 1-3. rewrite each vector system as a system of first-order DEs.1.2. 3.
Verification Verify that the vector functions , , and in Example 9 satisfy the IVP.
Third·Order Verification Verify that the vector functions . , and of Example 8 are indeed solutions of system (8), and that they are linearly independent.
Set up an Euler's method spreadsheet for each of the following systems:(a) Example 5 (b) Example 6 (c) Example 9 Choose an initial condition that corresponds to Fig. 6. 1. l, 6.1.2, or 6. I. 7 as appropriate. use a step size h = 0. 1, and confirm that the solutions indeed set off in the directions
For x as a function fo y. Use one or both of these techniques to determine and sketch phase-plane trajectories for the systems of problems a and b predict how the speed of a point trav-eling along a trajectory will be affected by the position of the point. a. x' = x y' = y b. x' = y v' = -x
For solutions of = A with A skew-symmetric, lhe lenglh of vector i: is constant. What does this mean in terms of phase-plane trajectories? Use IDE's Matrix Element Input tool, or an open-ended graphic DE solver, to verify this for the following systems, and explain lhe role of the parameter
In Problems 5-7, verify that the given vector functions satisfy the system; then give a fundamental matrix X(t) and the general solution (t).1.2.3.
Illustrate the graphical implications that trajectories can cross in t x and t y space, but not in the phase plane, by drawing phase-plane trajectories and component graphs for the autonomous systemExplain how different component solution graphs can have the same phase-plane traj ectory, by marking
In Problems 1-3,find the constant solution(s) x(t) = kfor each second-order equation, anddetermine the behiJvior as follows:(a) Rewrite the equation as a system of two first-order equations.(b) Find the equilibrium solution(s) of the equivalent first-order system.(c) Deduce the behavior of the
Find the general solutions for Problems a and b. Sketch the eigenvectors and a few typical trajectories. (Show your method.)a.b.
Solve the JVPs in Problems a - c. Sketch the trajectory.a.b. c.
Creating New Problems(b) Find a 3 x 3 matrix with a triple eigenvalue and two linearly independent eigenvectors.
Suppose that(a) Show that the system has a double eigenvalue if and only if the condition (a - d)2 + 4bc = 0 is satisfied, and that the eigenvalue is 1/2 (a + d) .(b) Show that if the condition in (a) holds and a = d, the eigenspace will be two-dimensional only if the matrixis diagonal.(c) Show
Suppose thal we wish to extend the method described for finding one generalized eigenvector to finding two (or more) generalized eigenvectors. Let's look at the case where has multiplicity 3 but has only one linearly independent eigenvector io. First. we find 1 by the method described in this
Consider for(a) Show that A has eigenvalue = 1 with multiplicity 3, and that all eigenvectors are scalar multiples of(b) Use part (a) to find a solution of the system in the fonn(c) Find a second solution in the formwhere vector is to be determined.(d) Use the result of part (c) to find a
Find the general solutions/or Problems a and b.a.b.
Verification of Independence Show that the solutions obtained in Example 7 of this section are linearly independent.
The linear systemhas a "cousin" systemcalled its adjoint. (Taking the negative of the transpose twice returns the original matrix, so each system is the adjoint of the other.)(c) Solve the IVP consisting of the system of part (a) and(d) Solve the IVP consisting of the adjoint of the system of
Cauchy-Euler Systems The systemwhere A is a constant matrix and t > 0.is called asystem.(a) Show tht the Cauchy-Euler system has a solution of the form where λ is an eigenvalue of A and is a corresponding eigenvector.(b) Solve the Cauchy-Euler system
Check your intuition for each of the systems in Problems a - c, following these steps. (a) Sketch what you think the direction.field looks like. (b) Use an open-ended solver to draw the vector field. (c) Solve the system analytically and compare with results in (a) and (b). Explain or reconcile any
The radioactive isotope of iodine, 1- 1 35, decays into the radioactive isotope Xe- 1 35 of xenon; this in tum decays into another (stable) product. The half-lives of iodine and xenon are 6.7 hours and 9.2 hours, respectively. (a) Write a system of differential equations descnbing the amounts of 1-
Consider two large tanks, connected as shown in Fig. 6.2. 15. Tank A is initially filled with I 00 gal of water in which 25 lb of salt has been dissolved. Tank B is initially filled with l 00 gal of pure water. Pure water is poured into tank A at the constant rate of 4gal/min. The well-mixed
Repeal Problem 52, but change the initial volume in tank A to 1 50 gal.
Why is the linear system that models the arrangement in Problem 52 homogeneous? Change the problem statement as simply as possible to keep the same matrix of coefficients A in the system for the new problem
A simple threecompartment model that describes nutrients in a food chain has been studied by M. R. Cullen.1 (See Fig. 6.2. 16.) For example, the constant a31 = 0.04 alongside the arrow connecting compartment I (phytoplankton) to compartment 3 (zooplankton) means that at any given time, nutrients
Detennine the genera] solutions for the cunents I1 . I2 and I3 if R1 = R2 = R3 = 4 ohms and L1 = L1 =2benries.
Find genera] solutions for the currents I1, I2 and I3 if R1 = 4 ohms, R3 = 6ohms, L1 = I henry, L2 = 2benries.
Find the general solutions for Problems a-c. Sketch the eigerrvectors and a few typical trajectories.(Show your method)a.b. c.
Find the general solutions for Problems a, and c handsketch the phase portraits from nullclines and/or vector fields. (You can check your results by using a CAS, a solver. or /DE Matrix Element Input.)a.b. c.
(a) Show that one of the nondiagonal elements, but not both, must be negative in order for the eigen values to be nonreal.(b) Show that eigenvalues are imaginary (Î± = 0) if and only if Tr A = 0 and |A| > 0.
rotation along trajectories is detennined as follows.€¢ if b is negative, rotation is counterclockwise;€¢ if c is negative, rotation is clockwise.
Elliptical phase-plane trajectories occur in a 2 x 2 DE system with purely imaginary eigenvalues, which happens when Tr A = 0 and |A| > 0. Thus, for (a) Show thatis an eigenvector for(b)Multjplying the vector in (a) by the complex scalar 1 / b(a + βi) gives a new vectorShow that is
For a 2 x 2 linear DE systemwe can make a quick handsketch of the elliptical trajectories as follows. (a) All the elliptical trajectories are concentric and similar, so we can get all the key information from just one. We have from equation (7) that By choosing the particular solution where c1 =I,
For Problems a - c(b) Add the appropriate velocity vectors at the end of each (See Problem 20) and rough-sketch the elliptical tra-jectory that passes through these four points. (c) Compare your sketch with a computer phase portrait from a graphic DE solver and explain any discrepan-cies. a. b. c.
For a system as described in Problem 20, we have the condition that at the ends of the major taonrd m m1i1snto r axes of an elliptical trajectory, the velocity vec· be orthogonal to the position vector. That is, we must haveUsing your equations from Problem 20(a), show that condition ( 19)
For Problems a - c(a) Using equation (20) from Problem 25, solve for the two values of {Jt€¢ for which ( 1 9) is satisfied.(b) Substitute (one at a time) each value of {Jt€¢ found in part (a) into the solutions found in Problem 20 (a) to calculate the endpoints of the major and

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