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Applied Linear Algebra 1st edition Peter J. Olver, Cheri Shakiban - Solutions

(a) In Example 5.62, the n = 4 discrete Fourier coefficients of the function f(x) = 2π c - x2 were found to be real. Is this true when n = 16? For general n? (b) What property of a function f(x) will guarantee that its Fourier coefficients are real?
Let c = (c0, c1,... , cn-1)T ˆŠ Cn be the vector of discrete Fourier coefficients corresponding to the sample vector f = (f0, f1,..., fn-1)T.(a) Explain why the sampled signal f = Fn c can be reconstructed by multiplying its Fourier coefficient vector by an n Ã— n matrix Fn.
A mass-spring chain consists of two masses connected to two fixed supports. The spring constants are c1 = c3 = 1 and c2 = 2. (a) Find the stiffness matrix K. (b) Solve the equilibrium equations K u = f when f = (4, 3)T. (c) Which mass moved the farthest? (d) Which spring has been stretched the
In a statically indeterminate situation, the equations ATy = f do not have a unique solution for the internal forces y in terms of the external forces f. (a) Prove that, nevertheless, if C = I. the internal forces are the unique solution of minimal Euclidean norm, as given by Theorem 5.59. (b) Use
Prove directly that the stiffness matrices in Examples 6.1 and 6.2 are positive definite.
Write down the potential energy for the following mass-spring chains with identical unit springs when subject to a uniform gravitational force: (a) Three identical masses connected to only a top support. (b) Four identical masses connected to top and bottom supports. (c) Four identical masses
(a) Find the total potential energy of the equilibrium configuration of the mass-spring chain in Exercise 6.1.1. (b) Test the minimum principle by substituting three other possible displacements of the masses and checking that they all have larger potential energy.
Answer Exercise 6.1.14 for the mass-spring chain in Exercise 6.1.4. (a) Determine the equilibrium positions of the masses and the elongations of the springs when the external force is f = (0, 1, 1, 0)T. Is your solution unique? (b) Suppose we only fix the top support. Solve the problem with the
Describe the mass-spring chains that gives rise to the following potential energy functions, and find their equilibrium configuration: (a) 3u21 - 4u1u2 + 3u22 + u1 - 3u2 (b) 5u21 - 6u1u2 + 3u22 + 2u2 (c) 2u21 - 3u1u2 + 4u22 - 5u2u23 - u1 - u2 + u3 (d) 2u21 - u1u2 + u22 - u2u3 + u23 - u3u4 + 2u24 +
Return to the situation investigated in Exercise 6.1.8. How should you arrange the springs in order to minimize the potential energy in the resulting mass-spring chain?
Solve Exercise 6.1.1 when the first and second spring are interchanged, c1 = 2, c2 = c3 = 1. Which of your conclusions changed? Exercise 6.1.2 (a) Find the stiffness matrix K. (b) Solve the equilibrium equations K u = f when f = (4, 3)T. (c) Which mass moved the farthest? (d) Which spring has been
True or false: The potential energy function uniquely determines the mass/spring chain.
Redo Exercises 6.1.1-2 when the bottom support and spring are removed. Exercise 6.1.1 A mass-spring chain consists of two masses connected to two fixed supports. The spring constants are c1 = c3 = 1 and c2 = 2. (a) Find the stiffness matrix K. (b) Solve the equilibrium equations K u = f when f =
A mass-spring chain consists of four masses suspended between two fixed supports. The spring stiffnesses are c1 = 1, c2 = 1/2, c3 = 2/3, c4 = 1/2, c5 = 1. (a) Determine the equilibrium positions of the masses and the elongations of the springs when the external force is f = (0, 1, 1, 0)T. Is your
(a) Show that, in a mass-spring chain with two fixed ends, under any external force, the average elongation of the springs is zero:(b) What can you say about the average elongation of the springs in a chain with one fixed end?
Suppose we subject the ith mass (and no others) in a chain to a unit force, and then measure the resulting displacement of the jth mass. Prove that this is the same as the displacement of the ith mass when the chain is subject to a unit force on the jth mass.
Find the displacements u1, u2, ... u100 of 100 masses connected in a row by identical springs, with spring constant c = 1. Consider the following three types of force functions: (a) Constant force: f1 = . . . = f100 = .01; (b) Linear force: fi = .0002i; (c) Quadratic force: fi = 6 ∙ 10-6 i (100 -
(a) Suppose you are given three springs with respective stiffnesses c = 1. cʹ = 2, cʹʹ = 3. In what order should you connect them to three masses and a top support so that the bottom mass goes down the farthest under a uniform gravitational force? (b) Answer Exercise 6.1.8 when the springs
Generalizing Exercise 6.1.8. suppose you are given n different springs. (a) In which order should you connect them to n masses and a top support so that the bottom mass goes down the farthest under a uniform gravitational force? Does your answer depend upon the relative sizes of the spring
Draw the electrical networks corresponding to the following incidence matrices.(a)(b) (c) (d) (e)
The nodes in an electrical network lie on the vertices (1/n, 1/n) for - n ≤ i, j ≤ n in a square grid centered at the origin: the wires run along the grid lines. The boundary nodes, when x or y = ± 1, are all grounded. A unit current source is introduced at the origin. (a) Compute the
Show that, in a network with all unit resistors, the currents y can be characterized as the unique solution to the Kirchhoff equations AT y = f of minimum Euclidean norm.
(a) Find the voltage potentials at all the nodes and the currents along the wires of the following trees if the bottom node is grounded and a unit current source is introduced at the top node.(i)(ii) (iii) (iv) (v) (b) Can you make any general predictions about electrical currents in trees?
A node in a tree is called terminating if it has only one edge. Repeat the preceding exercise when all terminating nodes except for the top one are grounded.(i)(ii) (iii) (iv) (v)
Suppose the graph of an electrical network is a tree, as in Exercise 2.6.8. Show that if one of the nodes in the tree is grounded, the system is statically determinate.
True or false: (a) The nodal voltage potentials in a network with batteries b are the same as in the same network with the current sources f = - AT C b. (b) Are the currents the same?
Suppose two wires in a network join the same pair of nodes. Explain why their effect on the rest of the network is the same as a single wire whose conductance c = c1 + c2 is the sum of the individual conductance's. How are the resistances related?
(a) Write down the equilibrium equations for a network that contains both batteries and current sources. (b) Formulate a general superposition principle for such situations.
Prove that the voltage potential at node i due to a unit current source at node j is the same as the voltage potential at node j due to a unit current source at node i. Can you give a physical explanation of this reciprocity relation?
What is the analog of condition (6.33) for a disconnected graph?
Suppose that all wires in the illustrated network have unit resistivity.(a) Write down the incidence matrix A. (b) Write down the equilibrium system for the network when node 4 is grounded and there is a current source of magnitude 3 at node 1. (c) Solve the system for the voltage potentials at the
What happens in the network in Figure 6.5 if we ground both nodes 3 and 4? Set up and solve the system and compare the currents for the two cases.
(a) Write down the incidence matrix A for the illustrated electrical network.(b) Suppose all the wires contain unit resistors, except for R4 = 2. Let there be a unit current source at node 1, and assume node 5 is grounded. Find the voltage potentials at the nodes and the currents through the
Answer Exercise 6.2.4 if, instead of the current source, you put a 1.5 volt battery on wire 1.Exercise 6.2.4(a) Write down the incidence matrix A for the illustrated electrical network.(b) Suppose all the wires contain unit resistors, except for R4 = 2. Let there be a unit current source at node 1,
(a) How do the currents change if the resistances in the wires in the cubical network in Example 6.4 are all equal to 1 ohm? (b) What if wire k has resistance Rk = k ohms?
Suppose you are given six resistors with respective resistances 1, 2, 3, 4, 5 and 6. How should you connect them in a tetrahedral network (one resistor per wire) so that a light bulb on the wire opposite the battery bums the brightest?
Consider a structure consisting of three bars joined in a vertical line hanging from a top support. (a) Write down the equilibrium equations for this system when only forces and displacements in the vertical direction are allowed, i.e., a onedimensional structure. Is the problem statically
A mass-spring ring consists of n masses connected in a circle by n identical springs, and the masses are only allowed to move in the angular direction. (a) Derive the equations of equilibrium. (b) Discuss stability, and characterize the external forces that will maintain equilibrium. (c) Find such
A space station is built in the shape of a three- dimensional simplex whose nodes are at the positions 0. e1, e2, e3 ∊ R3, and each pair of nodes is connected by a bar. (a) Sketch the space station and find its incidence matrix A. (b) Show that ker A is six-dimensional, and find a basis. (c)
Suppose a space station is built in the shape of a regular tetrahedron with all sides of unit length. Answer all questions in Exercise 6.3.12.(a) Sketch the space station and find its incidence matrix A.(b) Show that ker A is six-dimensional, and find a basis.(c) Explain which three basis vectors
True or false: If a structure is statically indeterminate, then every non-zero applied force will result in (a) One or more nodes having a non-zero displacement; (b) One or more bars having a non-zero elongation.
A structure in R3 has n movable nodes, admits no rigid motions, and is statically determinate. (a) How many bars must it have? (b) Find an example with n = 3.
Prove that if we apply a unit force to node i in a structure and measure the displacement of node j in the direction of the force, then we obtain the same value if we apply the force to node j and measure the displacement at node i in the same direction.
True or false: A structure in R3 will admit no rigid motions if and only if at least 3 nodes are fixed.
Let A be the reduced incidence matrix for a structure and C the diagonal bar stiffness matrix. Suppose f is a set of external forces that maintains equilibrium of the structure. (a) Prove that f = ATCg for some g. (b) Prove that an allowable displacement u is a least squares solution to the system
Suppose an unstable structure admits no rigid motions-only mechanisms. Let f be an external force on the structure that maintains equilibrium. Suppose that you stabilize the structure by adding in the minimal number of reinforcing bars. Prove that the given force f induces the same stresses in the
When a node is fixed to a roller, it is only permitted to move along a straight line-the direction of the roller. Consider the three bar structure in Example 6.5. Suppose node 1 is fixed, but node 4 is attached to a roller that only permits it to move in the horizontal direction. (a) Construct the
Answer Exercise 6.3.22 when the roller at node 4 only allows it to move in the vertical direction. Exercise 6.3.22 (a) Construct the reduced incidence matrix and the equilibrium equations in this situation. You should have a system of 5 equations in 5 unknowns-the horizontal and vertical
Redo Exercises 6.3.22-23 for the reinforced structure in Figure 6.16.(a) Construct the reduced incidence matrix and the equilibrium equations in this situation. You should have a system of 5 equations in 5 unknowns-the horizontal and vertical displacements of nodes 2 and 3 and the horizontal
(a) Suppose that we fix one node in a planar structure and put a second node on a roller. Does the structure admit any rigid motions? (b) How many rollers are needed to prevent all rigid motions in a three-dimensional structure? Are there any restrictions on the directions of the rollers?
(a) For the reinforced structure illustrated in Figure 6.16, determine the displacements of the nodes and the stresses in the bars under a uniform horizontal force, and interpret physically.(b) Answer the same question for the doubly reinforced structure in Figure 6.17.
Discuss the effect of a uniform horizontal force in the direction of the horizontal bar on the swing set and its reinforced version in Example 6.9.
All the bars in the illustrated square planar structure have unit stiffness.(a) Write down the reduced incidence matrix A.(b) Write down the equilibrium equations for the structure when subjected to external forces at the free nodes.(c) Is the structure stable? statically determinate? Explain in
In the square structure of Exercise 6.3.5, the diagonal struts simply cross each other. We could also try joining them at an additional central node. Compare the stresses in the two structures under a uniform horizontal and a uniform vertical force at the two upper nodes, and discuss what you
Write down the reduced incidence matrix A* for the pictured structure with 4 bars and 2 fixed supports. The width and the height of the vertical sides is I unit, while the top node is 1.5 units above the base.(a) Predict the number of independent solutions to A* u = 0, and then solve to describe
Consider the two-dimensional "house" constructed out of bars, as in the accompanying picture. The bottom nodes are fixed. The width of the house is 3 units, the height of the vertical sides 1 unit, and the peak is 1.5 units above the base.(a) Determine the reduced incidence matrix A for this
Answer Exercise 6.3.8 for the illustrated two- and three-dimensional houses. In the two-dimensional case, the width and total height of the vertical bars is 2 units, and the peak is an additional .5 unit higher. In the three-dimensional house, the width and vertical heights are equal to 1 unit, the
Let a = (a. b, c)T ∈ R3 be a fixed vector. Prove that the cross product map La[v] = a × v, as defined in (5.2), is linear, and find its matrix representative.
Let V be a vector space. Prove that every linear function L: R → V has the form L[x] = x b for x ∈ R, where b ∈ V is a fixed vector.
Let be fixed 2 Ã— 2 matrices. For each of the following functions, prove that L: M2Ã—2 †’ M2Ã—2 defines a linear map, and then find its matrix representative with respect to the standard basisof M2Ã—2. (a) L[X] = AX (b) R[AT] = XB (c) K[X] =
The domain space of the following functions is the space of n × n real matrices A. Which are linear? What is the target space in each case? (a) L[A] = 3A (b) L[A] = I - A (c) L[A] = AT (d) L[A] = A-l (e) L[A] = detA (f) L[A] = tr A (g) L[A] = diag A, i.e., the diagonal entries of A (h) L[A] = A v
(a) Prove that L is linear if and only if it satisfies (7-3). (b) Use induction to prove that L satisfies (7.4).
Let v1,... , vn be a basis of V and w1,... , wn be any vectors in W. Show that there is a unique linear function L: V → W such that L[v1] = wi, i = 1,... ,n.
Let V, W, Z be vector spaces. A function that takes any pair of vectors v ∈ V and w ∈ W to a vector z = B(v, w) ∈ Z is called bilinear if, for each fixed w, it is a linear function of v, so B(cv + d, w) = c B(v, w) + d B(, w), and, for each fixed v, it is a linear function of w, so B(v, cw
Which of the following define linear operators on the vector space C1(R) of continuously differentiable scalar functions? What is the target space?(a) L[f] = f(0) + f(1)(b) L[f] = f(0) f(1)(c) L[f1 = f€²(1)(d) L[f] = f€²(3) - f(2)(e) L[f] = x2 f(x)(f) L[f] = f(x + 2)(g) L[f] =
True or false: The average or meanof a function on the interval [a, b] defines a linear operator A : C0[a, b] †’ R.
Prove that multiplication Mh[f(x)] = h(x) f(x) by a fixed function h ∈ Cn[a, b] defines a linear operator Mh: Cn[a, b] → Cn[a, b]. Which result from calculus do you need to complete the proof?
Show that if w(x) is any fixed continuous function, then the weighted integraldefines a linear operator Iw,: C0[a, b] †’ R.
(a) Show that the partial derivativesboth define linear operators on the space of continuously differentiable functions f(x, y). (b) For which values of a, b, c, d is the map linear?
Prove that the Laplacian operatordefines a linear function on the vector space of twice continuously differentiable functions f(x, y).
Show that the gradient G[f] = ∇f defines a linear operator from the space of continuously differentiable scalar-valued functions f: R2 → R to the space of continuous vector fields v: R2 → R2.
Prove that, on R3, the gradient, curl and divergence all define linear operators. Be precise in your description of the domain space and the target space in each case.
Write down a basis for and dimension of the linear function spaces (a) L(R3, R) (b) L(R2, R2) (c) L(Rm, Rn) (d) L(P(3), R) (e) L(P(2), R2) (f) L(P(2), P(2)) Here P(n) is the space of polynomials of degree ≤ n.
Explain why the following functions F: R2 †’ R2 are not linear.(a)(b) (c) (d) (e)
Consider the linear function L: R3 †’ R defined by L(x, y, z) = 3x - y + 2z. Write down the vector a ˆˆ R3 such that L[v] = (a, v) when the inner product is(a) The Euclidean dot product,(b) The weighted inner product(v, w) = v1w1 + 2 v2w2 + 3 v3 w3,(c) The inner product defined
Let Rn be equipped with the inner product (v, w) = vTK w. Let L[v] = rv where r is a row vector of size 1 Ã— n.(a) Find a formula for the column vector a such that (7.12) holds for the linear function L: Rn †’ R.(b) Illustrate your result when r = (2, -1), using(i) The dot
Dual Bases: Given a basis v1,..., vn of V, the dual basis L1,... , Ln of V* consists of the linear functions uniquely defined by the requirements(a) Show that Li[v] = xi gives the ith coordinate of a vector v = x1v1 + ... + xnvn with respect to the given basis. (b) Prove that the dual basis is
Use Exercise 7.1.32(c) to find the dual basis for:(a)(b) (c) (d) (e) Exercise 7.1.32(c) Prove that if V = Rn and A = (v1 v2 ... vn) is the n Ã— n matrix whose columns are the basis vectors, then the rows of the inverse matrix A-1 can be identified as the corresponding dual basis of
Let P(2) denote the space of quadratic polynomials equipped with the L2 inner productFind the polynomial q that represents the following linear functions, i.e., such that L[p] = (q, p): (a) L[p] = p(0) (b) L[p] = 1/2p€²(1) (c) (d)
Write out a proof of Theorem 7.10 that does not rely on finding an orthonormal basis.
For each of the following pairs of linear functions S, T: R2 → R2, describe the compositions S ○ T and T ○ S. Do the functions commute?(a) S = counterclockwise rotation by 60°;T = clockwise rotation by 120°(b) S = reflection in the line y = x;T = rotation by 180°(c) S = reflection in the x
Find a matrix representative for the linear functions(a) L: R2 †’ R2 that maps ei toand e2 to (b) M: R2 †’ R2 that takes e1 to and e2 to and (c) N: R2 †’ R2 that takes and (d) Explain why M = N —‹ L. (e) Verify part (d) by multiplying the matrix
On the vector space R3, let R denote counterclockwise rotation around the x axis by 90° and S counterclockwise rotation around the z axis by 90°. (a) Find matrix representatives for R and S. (b) Show that R ○ S ≠ S ○ R. Explain what happens to the standard basis vectors under the two
Let P denote orthogonal projection of R3 onto the plane V = {z = x + y) and Q denote orthogonal projection onto the plane W = {z = x - y). Is the composition R = Q ○ P the same as orthogonal projection onto the line L = V ⋂ W? Verily your conclusion by computing the matrix representatives of P,
(a) Write the linear operator L[f (x)] = f′(b) as a composition of two linear functions. Do your linear functions commute? (b) For which values of a, b, c, d, e is L[f(x)] = a f′(b) + c f(d) + e a linear function?
Let L = x D + 1, and M = D - x be differential operators. Find L ○ M and M ○ L. Do the differential operators commute?
(a) Explain why the differential operator L = D ○ Ma ○ D obtained by composing the linear operators of differentiation D[f(x)] = f′(x) and multiplication Ma[f(x)] = a(x) f(x) by a fixed function a(x) defines a linear operator. (b) Re-express L as a linear differential operator of the form
(a) Show that composition of linear functions is associative: (L ○ M) ○ N = L ○ (M ○ N). Be precise about the domain and target spaces involved. (b) How do you know the result is a linear function? (c) Explain why this proves associativity of matrix multiplication.
Show that the space of constant coefficient linear differential operators of order ≤ n forms a vector space. Determine its dimension by exhibiting a basis.
Show that if p(x, y) is any polynomial, then L = p(ˆ‚x, ˆ‚y) defines a linear, constant coefficient partial differential operator. For example, if p(x, y) = x2 + y2, then L = ˆ‚x2 + ˆ‚y2 is the Laplacian operator
The commutator of two linear transformations L, M: V †’ V on a vector space V is defined asK = [L, M) = L —‹ M - M —‹ L. (7.15)(a) Prove that the commutator K is a linear transformation.(b) Prove that L and M commute if and only if [L, M] = 0.(c) Compute the
(a) In (one-dimensional) quantum mechanics, the differentiation operator P[f(x)] = f′(x) represents the momentum of a particle, while the operator Q[f(x)] = x f(x) of multiplication by the function x represents its position. Prove that the position and momentum operators satisfy the Heisenberg
Let D(1) denote the set of all first order linear differential operators L = p(x) D + q(x) where p, q are polynomials. (a) Prove that D(1) is a vector space. Is it finitedimensional or infinite-dimensional? (b) Prove that the commutator (7.15) of L, M ∈ D(1) is a first order differential operator
Find a matrix representation for the following linear transformations on R3: (a) A counterclockwise rotation by 90° around the 2-axis. (b) A clockwise rotation by 60° around the x-axis. (c) Reflection through the (x, y)-plane. (d) Counterclockwise rotation by 120° around the line x = y = z. (e)
Do the conclusions of Exercise 7.1.49(a-b) hold for the space D(2) of second order differential operators L = p(x) D2 + q(x) D + r(x) where p, q, r are polynomials?
Determine which of the following linear functions L: R2 †’ R2 has an inverse, and, if so, describe it:(a) The scaling transformation that doubles the length of each vector;(b) Clockwise rotation by 45°;(c) Reflection through the y axis;(d) Orthogonal projection onto the line y =
For each of the linear functions in Exercise 7.1.51, write down its matrix representative, the matrix representative of its inverse, and verify that the matrices are mutual inverses.Exercise 7.1.51Determine which of the following linear functions L: R2 †’ R2 has an inverse, and, if so,
Let L: R2 → R2 be the linear function such that L[e1] = (1, -1)T, L[e2] = (3, -2)T. Find L-1[e1] and L-1[t2].
Let L: R3 → R3 be the linear function such that L[e1] = (2, 1, -1)T, L[e2] = (1, 2, l)T, L[e3] = (-1, 2, 2)T. Find L-1[e1], L-1[e2] and L-1[e3].
Prove that the inverse of a linear transformation is unique; i.e., given L, there is at most one linear transformation M that can satisfy (7.17).

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