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

Suppose v1,..., vn is a basis for V and w1,..., wn a basis for W.(a) Prove that there is a unique linear function L: V †’ W such that L[vi] = wi for i = 1,..., n.(b) Prove that L is invertible.(c) If V = W = Rn, find a formula for the matrix representative of the linear functions L and L-1.(d)
Suppose V, W ⊂ Rn are subspaces of the same dimension. Prove that there is an invertible linear function L: Rn → Rn that takes V to W.
Give an example of a matrix with a left inverse, but not a right inverse. Is your left inverse unique?
Find a linear function L: R2 †’ R such thatIs it unique?
(a) Prove that L[p] = p′ + p defines an invertible linear map on the space P(2) of quadratic polynomials. Find a formula for its inverse. (b) Does the derivative D[p] = p′ have either a left or a right inverse on P(2)?
(a) Show that the set of all functions of the form f(x) = (ax2 + bx + c) ex for a, b, c, ∈ R forms a vector space. What is its dimension? (b) Show that the derivative D[f(x)] = f′(x) defines an invertible linear transformation on this vector space, and determine its inverse. (c) Generalize your
Under what conditions does there exist a linear function L: R2 †’ R2 such thatUnder what conditions is L uniquely defined? In the latter case, write down the matrix representation of L.
Can you construct a linear function L: R3 †’ R such thatand If yes, find one. If not, explain why not.
For each of the following linear transformations L: R2 → R2, find a matrix representative, and then describe its effect on(i) The x-axis;(ii) The unit square S = (0 ≤ x, v ≤ 1):(iii) The unit disk D = [x2 + y2 ≤ 1).(a) Counterclockwise rotation by 45°.(b) Rotation by 180°;(c) Reflection
(a) Prove that the linear transformation associated with the improper orthogonal matrixis a reflection through the line that makes an angle 1/2 Î¸ with the x-axis. (b) Show that the composition of two such reflections, with angles Î¸, Ï†, is a rotation. What is the
Let L ⊂ R2 be a line through the origin in the direction of the unit vector u. (a) Prove that the matrix representative of reflection through L is R = 2uuT - 1. (b) Find the corresponding formula for reflection through a line in the direction of a general nonzero vector v ≠ 0. (c) Determine the
Decompose the following matrices into a product of elementary matrices. Then interpret each of the factors as a linear transformation.
(a) Prove thatwhere a = -tan 1/2 Î¸ and b = sin Î¸. (b) Is the factorization valid for all values of Î¸? (c) Interpret the factorization geometrically.
(a) Find the matrix in R3 that corresponds to a counterclockwise rotation around the x-axis through an angle 60°. (b) Write it as a product of elementary matrices, and interpret each of the factors.
Determine the matrix representative for orthogonal projection P: R2 → R2 on the line through the origin in the direction (a) (1, 0)T (b) (1, 1)T (c) (2, -3)T
(a) Prove that any 2 × 2 matrix of rank 1 can be written in the form A = uvT where u, v ∈ R2 are non-zero column vectors. (b) Which rank one matrices correspond to orthogonal projection onto a one-dimensional subspace of R2?
Give a geometrical interpretation of the linear transformations on R3 defined by each of the six 3 × 3 permutation matrices (1.30).
Explain why the linear map defined by - I defines a rotation in two-dimensional space, but a reflection in three-dimensional space.
Draw the parallelogram spanned by the vectorsThen draw its image under the linear transformations defined by the following matrices:
Let u = (u1, u2, u3)T ∈ R3 be a unit vector. Show that Qn = 2uuT - I represents rotation around the axis u through an angle π.
Let u ˆˆ R3 be a unit vector.(a) Explain why the elementary reflection matrix R = I - 2uuT represents a reflection through the plane orthogonal to u.(b) Prove that R is an orthogonal matrix. Is it proper or improper?(c) Write out R when u =(d) Give a geometrical explanation why Qn = - R
Let a ∈ R3 be a fixed vector and let Q be any 3 × 3 rotation matrix such that Qa = e3. (a) Show that, using the notation of (7.23), Rθ = QT Zθ Q represents rotation around a by angle θ. (b) Verify this formula in the case a = e2 by comparing with (7.24).
Quaternions: The skew field H of quaternions can be identified with the vector space R4 equipped with a noncornmutative multiplication operation. The standard basis vectors e1, e2, e3, e4 are traditionally denoted by the letters 1, i, j, k; the vector (a, b, c, d)T ∈ R4 corresponds to the
Find the matrix form of the linear transformationwith respect to the following bases of R2:
Find the matrix form ofwith respect to the following bases of R3: (a) (b) (c)
Find bases of the domain and target spaces that place the following matrices in the canonical form (7.30). Use (7.29) to check your answer.
(a) Show that every invertible linear function L: Rn †’ Rn can be represented by the identity matrix by choosing appropriate (and not necessarily the same) bases on the domain and target spaces.(b) Which linear transformations are represented by the identity matrix when the domain and
Suppose A is an m × n matrix. (a) Let v1,..., vn be a basis of Rn, and Av1 = w1 ∈ Rm, for i = 1........n. Prove that the vectors v1,..., vn, w1,..., wn, serve to uniquely specify A. (b) Write down a formula for A.
Suppose a linear transformation L: Rn → Rn is represented by a symmetric matrix with respect to the standard basis e1,..., en. (a) Prove that its matrix representative with respect to any orthonormal basis u1,..., un is symmetric. (b) Is it symmetric when expressed in terms of a non-orthonormal
Let L be the linear transformation represented by the matrixShow that L2 = L —‹ L is rotation by 180°. Is L itself a rotation or a reflection?
In this exercise, we show that any inner product (ˆ™, ˆ™) on Rn can be reduced to the dot product when expressed in a suitably adapted basis.(a) Specifically, prove that there exists a basis v1,..., v" of Rn such thatwhere c = (c1, c2,... , cn)T are the coordinates of x and d =
Let L be the linear transformation determined byShow L2 = I, and interpret geometrically.
What is the geometric interpretation of the linear transformation with matrixUse this to explain why A2 = I.
Find a linear transformation that maps the unit circle x2 + y2 = 1 to the ellipse 1/4x2 + 1/9y2 = 1. Is your answer unique?
True or false: A linear transformation L: R2 → R2 maps (a) Straight lines to straight lines (b) Triangles to triangles (c) Squares to squares (d) Circles to circles (e) Ellipses to ellipses
Describe the image of(i) The x-axis(ii) The unit disk x2 + y2 ‰¤ 1(iii) The unit square 0 ‰¤ x, y ‰¤ 1under the following affine transformations:
Prove that the planar affine isometryrepresents a rotation through an angle of 90° around the center (3/2, -1/2)T.
Prove that every proper affine plane isometry F[x] = Qx + b of R2, where det Q = 1, is either (a) A translation, or (b) A rotation (7.40) centered at some point c ∈ R2.
Compute both compositions F ○ G and G ○ F of the following affine functions on R2. Which pairs commute? (a) F = counterclockwise rotation around the origin by 45°; G = translation in the y direction by 3 units. (b) F = counterclockwise rotation around the point (1, 1)T by 30°; G =
In R2, show the following: (a) The composition of two affine isometries is another affine isometry. (b) The composition of two translations is another translation. (c) The composition of a translation and a rotation (not necessarily centered at the origin) in either order is a rotation. (d) The
Let „“ be a line in R2. A glide reflection is an affine map on R2 composed of a translation in the direction of „“ by a distance d followed by a reflection through „“.Find the formula for a glide reflection along (a) The x axis by a distance 2, (b) The line y = x by a
Let ℓ be the line in the direction of the unit vector u through the point a. (a) Write down the formula for the affine map defining the reflection through the line ℓ. (b) Write down the formula for the glide reflection, as defined in Exercise 7.3.16, along ℓ by a distance d in the direction
A set of n + 1 points a0,..., an ∈ Rn is said to be in general position if the differences ai - aj span Rn. (a) Show that the points are in general position if and only if they do not all lie in a proper affine subspace A ⊊ Rn, cf. Exercise 2.2.30. (b) Let a0,..., an and b0,..., bn be two sets
Suppose that V is an inner product space and L: V → V is an isometry, so ||L[v]|| = ||v|| for all v ∈ V. Prove that L also preserves the inner product: (L[v], L[w]) = (v, w).
Using the affine functions in Exercise 7.3.1, write out the following compositions and verify that they satisfy (7.34): (a) T3 o T4 (b) T4 o T3 (c) T3 o T6 (d) T6 o T3 (e) T7 o T8 (f) T8 o T7
Let V be a normed vector space. Prove that a linear map L: V → V defines an isometry of V for the given norm if and only if it maps the unit sphere S1 = {||u|| = 1} to itself: L[S1] = {L[u] | u ∈ S1} = S1.
(a) List all linear and affine isometries of R2 with respect to the ∞ norm. (b) Can you generalize your results to R3?
Answer Exercise 7.3.21 for the 1 norm. Exercise 7.3.21 (a) List all linear and affine isometries of R2 with respect to the ∞ norm. (b) Can you generalize your results to R3?
Let „“ Š‚ R2 be a line, and p ˆ‰ „“ a point. A perspective map takes a point x ˆˆ R2 to the point q ˆˆ „“ that is the intersection of „“ with the line going through p and x. If the line is parallel to „“ then
True or false: An affine transformation takes (a) Straight lines to straight lines (b) Triangles to triangles (c) Squares to squares (d) Circles to circles (e) Ellipses to ellipses
Under what conditions is the composition of two affine functions (a) A translation? (b) A linear function?
(a) Under what conditions does an affine function have an inverse? (b) Is the inverse an affine function? If so, find a formula for its matrix and vector constituents. (c) Find the inverse, when it exists, of the affine functions in Exercise 7.3.1.
Let v1,..., vn be a basis for Rn. (a) Show that any affine function F[x] = Ax + b on Rn is uniquely determined by the n + 1 vectors w0 = F[0], w1 = F[v1],..., wn = F[vn]. (b) Find the formula for A and b when v1 = e1,..., vn = en are the standard basis vectors. (c) Find the formula for A. b for a
Show that the space of all affine functions on Rn forms a vector space. What is its dimension?
In this exercise, we establish a useful matrix representation for affine functions. We identify Rn with the n-dimensional affine subspace (as in Exercise 2.2.30)consisting of vectors whose last coordinate is fixed at xn+1 = 1. (a) Show that multiplication of vectors by the (n + 1) Ã— (n
Place each of the following linear systems in the form (7.42). Carefully describe the linear function, its domain space, its target space, and the right hand side of the system. Which systems are homogeneous?(a) 3x + 5 = 0(b) x = y + z(c) a = 2b - 3, b = c - 1(d) 3(p - 2) = 2(q - 3), p + q = 0(e)
The following functions are solutions to a constant coefficient homogeneous scalar ordinary differential equation. (i) Determine the least possible order of the differential equation, and (ii) Write down an appropriate differential equation. (a) e2x + e-3x (b) 1 + e-x (c) xex (d) ex + 2e2x + 3e3x
Solve the following Euler differential equations:(a) x2u" + 5xu' - 5u = 0(b) 2x2u" - xu' - 2u = 0(c) x2u" - u = 0(d) x2u" + xu' - 3u = 0(e) 3x2u" - 5xu' - 3u = 0(f)
Solve the third order Euler differential equation x3u"' + 2x2u" - 3xu' + 3u = 0 by using the power ansatz (7.52). What is the dimension of the solution space for x > 0? For all x?
(i) Show that if u(x) solves the Euler equationthen v(t) = u(et) solves a linear, constant coefficient differential equation. (ii) Use this alternative technique to solve the Euler differential equations in Exercise 7.4.11. Exercise 7.4.11 (a) x2u" + 5xu' - 5u = 0 (b) 2x2u" - xu' - 2u = 0 (c) x2u"
(a) Use the method in Exercise 7.4.13 to solve an Euler equation whose characteristic equation has a double root r1 = r2 = r. Solve the specific equations(i) x2u" - xu€² + u = 0(ii)
Show that if u (x) solves xu" + 2u′ - 4xu = 0, then v(x) = xu(x) solves a linear, constant coefficient equation. Use this to find the general solution to the given differential equation. Which of your solutions are continuous at the singular point x = 0? Differentiable?
Let S ⊂ R be an open subset (i.e., a union of open intervals). True or false: ker D is a one-dimensional subspace of C1 (S).
Show that log(x2 + y2) and x/x2 + y2 are harmonic functions, that is, solutions of the two-dimensional Laplace equation.
The Fredholm Alternative of Theorem 5.55 first appeared in the study of what are now known as Fredholm integral equations:in which K(x, y) and f(x) are prescribed continuous functions. Explain how the integral equation is a linear system; i.e., describe the linear map L, its domain and target
(a) Show that the function ex cos y is a solution to the two-dimensional Laplace equation. (b) Show that its quadratic Taylor polynomial at x = y = 0 is harmonic. (c) What about its degree 3 Taylor polynomial? (d) Can you state a general theorem? (e) Test your result by looking at the Taylor
(a) Find a basis for, and the dimension of the vector space consisting of all quadratic polynomial solutions of the three-dimensional Laplace equation(b) Do the same for the homogeneous cubic polynomial solutions.
Let L, M be linear functions. (a) Prove that ker(L ○ M) ⊇ ker M. (b) Find an example where ker(L ○ M) ≠ ker M.
For each of the following inhomogeneous systems, determine whether the right hand side lies in the range of the coefficient matrix, and, if so, write out the general solution, clearly identifying the particular solution and the kernel element.
Which of the following systems has a unique solution?(a)(b) (c) (d) (e)
Solve the following inhomogeneous linear ordinary differential equations: (a) u′ - 4u = x - 3 (b) 5u" - 4u′ + 4u = ex cos a, (c) u" - 3u′ = e3x
Solve the following initial value problems: (a) u' + 3u = ex, u(1) = 0 (b) u" + 4u = 1, u(π) = u'(π) = 0 (c) u" - u' - 2u = ex + e-x, u(0) = u'(0) = 0 (d) u" + 2u′ + 5u = sin x, u(0) = 1, u'(0) = 0 (e) u‴ - u" + u' - u = x, u(0) = 0, u'(0) = 1, u"(0) = 0
Write down all solutions to the following boundary value problems. Label your answer as (i) Unique solution, (ii) No solution, (iii) Infinitely many solutions. (a) u" + 2u = 2a, u(0) = 0, u(π) = 0 (b) u" + 4u = cos x, u(-π) = 0, u(π) = 1 (c) u" - 2u' + u = x - 2, u(0) = -1, u(1) = 1 (d) u" + 2u'
Solve the following inhomogeneous Euler equations using either variation of parameters or the change of variables method discussed in Exercise 7.4.13: (a) x2u" + xu' - u = x (b) x2u" - 2xu' + 2u = log x (c) x2u" - 3xu' - 5u = 3x - 5
Answer Exercise 7.4.2 for the Volterra integral equationwhere a ‰¤ t ‰¤ b. Exercise 7.4.2 The Fredholm Alternative of Theorem 5.55 first appeared in the study of what are now known as Fredholm integral equations: in which K(x, y) and f(x) are prescribed continuous functions.
Let L, M be linear functions. (a) Prove that rng(L ○ M) ⊆ rng L. (b) Give an example where rng(L ○ M) ≠ rng L.
Let L: U → V be a linear function, and let W ⊂ U be a subspace of the domain space. (a) Prove that Y = {L[w] | w ∈ W} ⊂ rng L ⊂ V is a subspace of the range. (b) Prove that dim Y ≤ dim W. Conclude that a linear transformation can never increase the dimension of a subspace.
Find all its invariant subspaces W ⊂ R2, as defined in Exercise 7.4.32, of the following linear transformations L: R2 → R2: (a) The scaling transformation (2x, 3y)T (b) The shear (x + 3y, y)T (c) Counterclockwise rotation by an angle 0 ≤ θ < 2π
(a) Show that if L: V → V is linear and ker L ≠ {0}, then L is not invertible. (b) Show that if rng L ≠ V, then L is not invertible. (c) Give an example of a linear map with ker L = {0} which is not invertible.
Use superposition to solve the following inhomogeneous ordinary differential equations: (a) u' + 2u = 1 + cos x (b) u" - 9u = x + sin x (c) 9u" - 18u' + 10u = 1 + ex cos x (d) u" + u' - 2u = sinh x, where Sinh x = 1/2 (ex - e-x) (e) u"' + 9u' = 1 + e3x
Solve the following boundary value problems by using superposition: (a) u" + 9u = x, u(0) = 1, u'(π) = 0 (b) u" - 8u′ + 16u = e4x, u(0) = 1, u(1) = 0 (c) u" + 4u = sin 3x, u'(0) = 0, u(2π) = 3 (d) u" - 2u' + u = 1 + ex, u'(0) = -1, u'(1) = 1
(a) Prove that the solution to the linear integral equationsolves the linear initial value problemdu/dt = k(t) u(t),u(0) = a.(b) Use part (a) to solve the following integral equations
Reduction of Order: Suppose you know one solution u1(x) to the second order homogeneous differential equation u" + a(x)u′ + b(x)u = 0. (a) Show that if u(x) = v(x) u1(x) is any other solution, then w(x) = v'(x) satisfies a first order differential equation. (b) Use reduction of order to find the
Write out the details of the proof of Theorem 7.43.
Can you find a complex matrix A such that ker A ≠ {0} and the real and imaginary parts of every complex solution to Au = 0 are also solutions?
Find the general real solution to the following homogeneous differential equations: (a) u" + 4u = 0 (b) u" + 6u' + 10u = 0 (c) 2u'" + 3u' - 5u = 0 (d) u"" + u = 0 (e) u"" + 13u" + 36u = 0 (f) x2 u" - xu′ + 3u = 0 (g) x3u'" + x2u" + 3xu' - 8u = 0
The following functions are solutions to a real constant coefficient homogeneous scalar ordinary differential equation. (i) Determine the least possible order of the differential equation, and (ii) Write down an appropriate differential equation. (a) e-x sin (b) 3x (c) xsin x (d) 1 + xe-x cos2x (e)
Find the general solution to the following complex ordinary differential equations. Verify that, in these cases, the real and imaginary parts of a complex solution are not real solutions. (a) u' + iu = 0 (b) u" - iu' + (i - 1)u = 0 (c) u" - iu = 0
(a) Write down the explicit formulas for the harmonic polynomials of degree 4 and check that they are indeed solutions to the Laplace equation. (b) Prove that every homogeneous polynomial solution of degree 4 is a linear combination of the two basic harmonic polynomials.
(a) Show that u(t, x) = e-k2t+ikx is a complex solution to the heat equation ∂u/∂t = ∂2u/∂x2 for any real constant k. (b) Write down another complex solution by using complex conjugation. (c) Find two independent real solutions to the heat equation. (d) Can k be complex? If so, what real
Find all complex exponential solutions u(t, x) = ewt+kx of the beam equation ∂2u/∂t2 = ∂4u/∂x4. How many different real solutions can you produce?
Prove that the real and imaginary parts of a general element of a conjugated vector space, as defined by (7.70), are both real elements.
Prove that a subspace V ⊂ Cn is conjugated if and only if it admits a basis all of whose elements are real.
Prove that a linear function L: Cn → Cm is real if and only if L[u] = Au where A is a real m × n matrix.
(a) Show that u solves the complex homogeneous linear system L[u] = 0 if and only if its complex conjugate v = solves the complex conjugate system [v] = 0.(b) Solveand then use your result to write down the solution to the system
Let u = x + iy be a complex solution to a real linear system. Under what conditions are its real and imaginary parts x, y linearly independent real solutions?
Solve the following homogeneous linear ordinary differential equations. What is the dimension of the solution space? (a) u" - 4u = 0 (b) u" - 6u' + 8u = 0 (c) u‴ - 9u′ = 0 (d) u"" + 4u‴ - u" - 16u′ - 12u = 0
Define L[y] = y" + y. (a) Prove directly from the definition that L: C2[a, b] → C0[a, b] is a linear transformation. (b) Determine ker L.
Answer Exercise 7.4.7 when L = 3D2 - 2D - 5. Exercise 7.4.7 Define L[y] = y" + y. (a) Prove directly from the definition that L: C2[a, b] → C0[a, b] is a linear transformation. (b) Determine ker L.
Consider the linear differential equation y'" + 5y" + 3y' - 9y = 0. (a) Write the equation in the form L[y] = 0 for a differential operator L = p(D). (b) Find a basis for kerL, and then write out the general solution to the differential equation.

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