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Linear Algebra A Modern Introduction 4th edition David Poole - Solutions

Let T: P2 †’ R2 be the linear transformation defined by(a) Which, if any, of the following polynomials are in ker(T)? (i) 1 + x (ii) x - x2 (iii) 1 + x - x2 (b) Which, if any, of the following vectors are in range(T)? (i) (ii) (iii) (c) Describe ker(T) and range(T).
(a) Show that b[0, l] = b[2, 3]. (b) Show that b[0, l] = ≅ b[a, a + l ] for all a.
Show that b[0, l] = b[0, 2].
Show that b[a, b] = b[c, d] for all a < b and c < d.
Let S : V → W and T: U → V be linear transformations. (a) Prove that if S and T are both one-to-one, so is S ͦ T. (b) Prove that if S and T are both onto, so is S ͦ T.
Let S: V → W and T: U → V be linear transformations. (a) Prove that if S ͦ T is one-to-one, so is T. (b) Prove that if S ͦ T is onto, so is S.
Let T: V → W be a linear transformation between two finite-dimensional vector spaces. (a) Prove that if dim V < dim W, then T cannot be onto. (b) Prove that if dim V > dim W, then T cannot be one-to-one.
Let a0, a1, . . . , an be n + 1 distinct real numbers. Define T: Pn †’ Rn+ 1 byProve that T is an isomorphism.
If V is a finite-dimensional vector space and T: V → V is a linear transformation such that rank(T) = rank(T2), prove that range(T)∩ ker(T) = {0}.
Let U and W be subspaces of a finite-dimensional vector space V. Define T: U × W → V by T(u, w) = u - w. (a) Prove that T is a linear transformation. (b) Show that range(T) = U + W. (c) Show that ker(T) = U ≅ W. (d) Prove Grassman n's Identity: dim (U + W) = dim U + dim W - dim( U ∩ W)
Let T: P2 → P2 be the linear transformation defined by T(p(x)) = xp'(x). (a) Which, if any, of the following polynomials are in ker( T)? (i) 1 (ii) x (iii) x2 (b) Which, if any, of the polynomials in part (a) are in range(T)? (c) Describe ker(T) and range(T).
In Exercises 1-3, find either the nullity or the rank of T and then use the Rank Theorem to find the other.1. T: M22 †’ R2 defined by2. T: P2 †’ R2 defined by 3. T: M22 †’ M22 defined by T(A) = AB, where
In Exercises 1-3, find the matrix [T]C←B of the linear transformation T: V → W with respect to the bases 13 and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem. 1. T: P1 → P1 defined by T(a + bx) = b - ax, B = C = {1, x}, v =
Consider the subspace W of D, given by W = span (sin x, cos x). (a) Show that the differential operator D maps W into itself. (b) Find the matrix of D with respect to B = {sin x, cos x}. (c) Compute the derivative of f(x) = 3 sin x - 5 cos x indirectly, using Theorem 6.26, and verify that it agrees
Consider the subspace W of D, given by W = span (e2x, e-2x). (a) Show that the differential operator D maps W into itself. (b) Find the matrix of D with respect to B = {e2X, e-2x}. (c) Compute the derivative of f(x) = e2x - 3e-zx indirectly, using Theorem 6.26, and verify that it agrees with f'(x)
Consider the subspace W of D, given by W = span (e2X, e2x cos x, e2x sin x). (a) Find the matrix of D with respect to B = {e2x, e2x cos x, e2x sin x}. (b) Compute the derivative of f(x) = 3e2x - e2x cos x + 2e2x sin x indirectly, using Theorem 6.26, and verify that it agrees with f' (x) as computed
Consider the subspace W of D, given by W = span (cos x, sin x, x cos x, x sin x). (a) Find the matrix of D with respect to B = {cos x, sin x, x cos x, x sin x}. (b) Compute the derivative of f(x) = cos x + 2x cos x indirectly, using Theorem 6.26, and verify that it agrees with f' (x) as computed
T: U †’ V and S: V †’ W are linear transformations and B, C, and D are bases for U, V, and W, respectively. Compute [S Í¦ T]D†B in two ways:(a) By finding S Í¦ T directly and then computing its matrix and(b) By finding the matrices of S and
In Exercises 1-3, determine whether the linear transformation T is invertible by considering its matrix with respect to the standard bases. If T is invertible, use Theorem 6.28 and the method of Example 6.82 to find T-1.1. T: P2 †’ P2 defined by T(p(x)) = p'(x)2. T: P2 †’ P2
In Exercises 1-2, use the method of Example 6.83 to evaluate the given integral. 1. ∫ (sin x - 3 cos x) dx. 2. ∫ 5e-2x dx.
In Exercises 1-2, a linear transformation T: V †’ V is given. If possible, find a basis C for V such that the matrix [T] c of T with respect to C is diagonal.1.2.
Let ‚¬ be the line through the origin in R2 with direction vectorUse the method of Example 6.85 to find the standard matrix of a reflection in „“.
Let W be the plane in R3 with equation x - y + 2z = 0. Use the method of Example 6.85 to find the standard matrix of an orthogonal projection onto W. Verify that your answer is correct by using it to compute the orthogonal projection of v onto W, whereCompare your answer with Example 5.11.
Let T: V → W be a linear transformation between finite-dimensional vector spaces and let B and C be bases for V and W, respectively. Show that the matrix of T with respect to B and C is unique. That is, if A is a matrix such that A [v]B = [T(v)]c for all v in V, then A = [T]C←B.
In Exercises 1-3, let T: V → W be a linear transformation between finite-dimensional vector spaces V and W Let B and C be bases for V and W, respectively, and let A = [T] C←B. 1. Show that nullity(T) = nullity(A). 2. Show that rank(T) = rank(A). 3. If V = W and B = C, show that T is
In Exercises 1-3, find the solution of the differential equation that satisfies the given boundary condition(s). 1. y' - 3y = 0, y (l) = 2 2. x' + x = 0, x (1) = 1 3. y" - 7y' + 12y = 0, y(0) = y (1) = 1
A strain of bacteria has a growth rate that is proportional to the size of the population. Initially, there are 100 bacteria; after 3 hours, there are 1600. (a) If p(t) denotes the number of bacteria after t hours, find a formula for p(t). (b) How long does it take for the population to double? (c)
Table 6.2 gives the population of the United States at 10-year intervals for the years 1900-2000. (a) Assuming an exponential growth model, use the data for 1900 and 1 9 1 0 to find a formula for p(t), the population in year t. (b) Repeat part (a), but use the data for the years 1970 and 1980 to
The half-life of radium-226 is 1S90 years. Suppose we start with a sample of radium-226 whose mass is SO mg. (a) Find a formula for the mass m(t) remaining after t years and use this formula to predict the mass remaining after 1000 years. (b) When will only 10 mg remain?
Radiocarbon dating is a method used by scientists to estimate the age of ancient objects that were once living matter, such as bone, leather, wood, or paper.All of these contain carbon, a proportion of which is carbon- 14, a radioactive isotope that is continuously being formed in the upper
A mass is attached to a spring, as in Example 6.92. At time t = 0 second, the spring is stretched to a length of 10 cm below its position at rest. The spring is released, and its length 10 seconds later is observed to be 5 cm. Find a formula for the length of the spring at time t seconds.Example
A 50 g mass is attached to a spring, as in Example 6.92. If the period of oscillation is 1 0 seconds, find the spring constantExample 6.92
A pendulum consists of a mass, called a bob, that is affixed to the end of a string of length L (see Figure 6.24). When the bob is moved from its rest position and released, it swings back and forth. The time it takes the pendulum to swing from its farthest right position to its farthest left
Show that the solution set S of the second-order differential equation y" + ay' + by = 0 is a subspace of P.
Show that eP1 cos qt and eP1 sin qt are linearly Independent
Mark each of the following statements true or false: (a) If V = span(v1, . . . , vn), then every spanning set for V contains at least n vectors. (b) If {u, v, w} is a linearly independent set of vectors, then so is {u + v, v + w, u + w}. (c) M22 has a basis consisting of invertible matrices. (d)
Find the change-of-basis matrices PC←B and PS←C with respect to the bases B = {1, 1 + x, 1 + x + x2} and C = {1 + x, x + x2, 1 + x2} of P2.
In Questions 1-3, determine whether T is a linear transformation.1. T: R2 †’ R2 defined by T(x) = yxTy, where2. T: Mnn †’ Mnn defined by T(A) = ATA 3. T: Pn †’ Pn defined by T(p(x)) = p(2x - 1)
If T: P2 †’ M22 is a linear transformation such thatAnd Find T(5 - 3x + 2x2).
Find the nullity of the linear transformation T: Mnn → R defined by T(A) = tr(A).
Let W be the vector space of upper triangular 2 × 2 matrices. (a) Find a linear transformation T: M22 → M22 such that ker(T) = W. (b) Find a linear transformation T: M22 → M22 such that range(T) = W.
Find the matrix [T]C←B of the linear transformation T in Question 14 with respect to the standard bases B = {l, x, x2} of P2 and C = {E11, E12, E21, E22} of M22.
Let S = {v1, . . . , vn} be a set of vectors in a vector space V with the property that every vector in V can be written as a linear combination of v1, . . . , vn in exactly one way. Prove that S is a basis for V.
In Questions 1-3, determine whether W is a subspace of V1.2. 3. V = P3, W = {p(x) in P3: x3 p(1 / x) = p(x)}
Let T: V → V be a linear transformation, and let {v1, . . . , vn} be a basis for V such that {T(v1), . . . , T (v)} is also a basis for V. Prove that T is invertible.
Determine whether {1, cos 2x, 3 sin2x} is linearly dependent or independent.
Let A and B be nonzero n × n matrices such that A is symmetric and B is skew-symmetric. Prove that {A, B} is linearly independent.
In Questions 1 and 2, find a basis for W and state the dimension of W1.2. W = {p (x) in P5: p(-x) = p(x)}
In Exercise 1 - 4, let1 . (u, v) is the inner product of Example 7.2. Compute (a) (u, v) (b) ||u|| (c) d(u, v) 2. (u, v) is the inner product of Example 7.3 with Compute (a) (u, v) (b) ||u|| (c) d(u, v) 3. In Exercise l, find a nonzero vector orthogonal to u. 4. In Exercise 2, find a nonzero vector
Let a, b, and c be distinct real numbers. Show that (p(x), q(x)) = p(a)q (a) + p(b)q(b) + p (c)q(c) defines an inner product on P2.
Repeat Exercise 5 using the inner product of Exercise 11 with a = 0, b = l, c = 2. In exercise 11 Let a, b, and c be distinct real numbers. Show that (p(x), q(x)) = p(a)q (a) + p(b)q(b) + p (c)q(c) defines an inner product on P2.
In Exercises 1- 3, determine which of the four inner product axioms do not hold. Give a specific example in each case.1. LetIn R2. Define (u, v) = u1v1. 2. Let In R2. Define (u, v) = u1v1 - u2v2. 3. Let In R2. Define (u, v) = u1v2 + u2v1.
In Exercises 1 and 2, (u, v) defines an inner product on R2, whereFind a symmetric matrix A such that (u, v) = uT Av. 1. (u, v) = 4u1v1 + u1v2 + u2v1 + 4u2v2 2. (u, v) = u1v1 + 2u1v2 + 2u2v1 + 5u2v2
In Exercises 1 and 2, sketch the unit circle in R2 for the given inner product, where1. (u, v) = u1v1 + 1/4u2v2 2. (u, v) = 4u1v1 + u1v2 + u2v1 + 4u2v2
In Exercises 1-3, suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 5, (v, w) = 0 ||u|| = 1, ||v|| = √3, ||w|| = 2 Evaluate the expressions in Exercises 25-28. 1. (u + w, v - w) 2. (2v - w, 3u + 2w) 3. ||u + v||
Show that, in an inner product space, there cannot be unit vectors u and v with (u, v) < - 1.
In Exercises 1-3, (u, v) is an inner product. In Exercises 31 -34, prove that the given statement is an identity.1. (u + v, u - v) = ||u||2 - ||v||22. ||u + v||2 = ||u||2 + 2(u, v) + ||v||23. ||u||2 + ||v||2 = 1/2 ||u + v||2 + 1/2 ||u - v||2
In Exercises 1-3, apply the Gram-Schmidt Process to the basis l3 to obtain an orthogonal basis for the inner product space V relative to the given inner product.1. V = R2,with the inner product in Example 7.2 2. V = R2, with the inner product immediately following Example 7.3 3. V = P2, B = {1, 1 +
(a) Compute the first three normalized Legendre polynomials. (See Example 7.8.) (b) Use the Gram-Schmidt Process to find the fourth normalized Legendre polynomial.
If we multiply the Legendre polynomial of degree n by an appropriate scalar we can obtain a polynomial Ln(x) such that Ln( l) = 1 .(a) Find L0(x), L1 (x), L2 (x), and L3(x).(b) It can be shown that L"(x) satisfies the recurrence Relationfor all n ‰¥ 2. Verify this recurrence for L2(x) and
Verify that if W is a subspace of an inner product space V and v is in V, then perpw (v) is orthogonal to all w in W.
Let u and v be vectors in an inner product space V. Prove the Cauchy-Schwarz Inequality for u ≠ 0 as follows: (a) Let t be a real scalar. Then (tu + v, tu + v) ≥ 0 for all values of t. Expand this inequality to obtain a quadratic inequality of the form at2 + bt + c ≥ 0 What are a, b, and c in
In Exercise 1 - 4, p(x) = 3 - 2x and q(x) = 1 + x + x2. 1. (p(x), q(x)) is the inner product of example 7.4. Compute (a) (p(X), q(x)) (b) ||p(x)|| (c) d(p(x), q(x)) 2. (p(x), q(x)) is the inner product of Example 7.5 on the vector P2 [0, 1]. Compute (a) (p(x), q(x)) (b) ||p(x)|| (c) d(p(x),
In Exercises 1 and 2, let f(x) = sin x and g(x) = sin x + cos x in the vector space l [0, 2π] with the inner product defined by Example 7.5. 1. Compute (a) f,g) (b) ||f|| (c) d(f, g) 2. Find a nonzero vector orthogonal to f.
In Exercise 1-3, let1. Compute the Euclidean norm, the sum norm, and the max norm of u. 2. Compute the Euclidean norm, the sum norm, and the max norm of v. 3. Compute d ( u, v) relative to the Euclidean norm, the sum norm, and the max norm.
Draw the unit circles in R2 relative to the sum norm and the max norm.
By showing that the identity of Exercise 33 in Section 7. 1 fails, show that the sum norm does not arise from any inner product.
In Exercises 15-18, prove that || || defines a norm on the vector space V.1.
In Exercises 1-3, compute ||A|F, ||A||1, and ||A||ˆž.1.2.3.
(a) If ||A|| is an operator norm, prove that ||I|| = 1, where I i s an identity matrix. (b) Is there a vector norm that induces the Frobenius norm as an operator norm? Why or why not?
In Exercises 35-40, find cond1 (A) and condˆž (A). State whether the given matrix is ill-conditioned.1.2. 3.
(a) What does ds (u, v) measure? (b) What does dm (u, v) measure?
(a) Find a formula for condˆž (A) in terms of k.(b) What happens to condˆž(A) as k approaches 1 ?Let
Consider the linear system Ax = b, where A is invertible. Suppose an error Δb changes b to b' = b + Δb. Let x' be the solution to the new system; that is, Ax' = b'. Let x' = x + Δx so that Δx represents the resulting error in the solution of the system. Show that for any compatible matrix
(a) Compute condˆž(A).(b) Suppose A is changed to
(a) Compute cond1(A).(b) Suppose A is changed to
Let A be an invertible matrix and let λ1 and λn be the eigenvalues with the largest and smallest absolute values, respectively. Show that cond (A) ≥ |λ1| / |λn|
In Exercises 1 and 2, let u = [1 0 1 1 0 0 1]T and v = [0 1 1 0 1 1 1]T. 1. Compute the Hamming norms of u and v. 2. Compute the Hamming distance between u and v.
Let A be an n × n matrix such that ||A|| < 1, where the norm is either the sum norm or the max norm. (a) Prove that An → O as n → ∞. (b) Deduce from (a) that I - A is invertible and (I - A)-1 = I + A + A2 + A3 + · · · (c) Show that (b) can be used to prove Corollaries 3.35 and 3.36.
(a) For which vectors v is ||v||E = ||v||m? Explain your answer. (b) For which vectors v is ||v||, = ||v||m? Explain your answer. (c) For which vectors v is ||v||, = ||v||m = ||v||E? Explain your answer.
(a) Under what conditions on u and v is ||u + v||E = ||u||E + ||v||E? Explain your answer. (b) Under what conditions on u and v is ||u +v||, = ||u||, + ||v||s? Explain your answer. (c) Under what conditions on u and v is ||u + v||m = ||u||m + ||v||m? Explain your answer.
1. Show that for all v in Rn, ||v||m ≤ ||v||E· 2. Show that for all v in Rn, ||v||E ≤ ||v||s. 3. Show that for all v in Rn, ||v||s ≤ n ||v||m·
In Exercises 1-3, consider the data points (1, 0), (2, 1), and (3, 5). Compute the least squares error for the given line. In each case, plot the points and the line. 1. y = - 2 + 2x 2. y = x 3. y = - 3 + 5/2 x
In Exercises 1-2, find the least squares approximating parabola for the given points. 1. (1, 1), (2, - 2), (3, 3), (4, 4) 2. (1, 6), (2, 0), (3, O), (4, 2)
In Exercises 1 9-22, find a least squares solution of Ax = b by constructing and solving the normal equations. 1. 2.
In Exercises 23 and 24, show that the least squares solution of Ax = b is not unique and solve the normal equations to find all the least squares solutions 1. 2.
In Exercises 25 and 26, find the best approximation to a solution of the given system of equations. 1. x + y - z = 2 - y + 2 z = 6 3x + 2y - z = 1 1 -x + z = 0 2. 2x + 3y + z = 2 1 x + y + z = 7 - x + y - z = 14 2y + z = 0
In Exercises 27 and 28, a QR factorization of A is given. Use it to find a least squares solution of Ax = b. 1. 2.
A tennis ball is dropped from various heights, and the height of the ball on the first bounce is measured. Use the data in Table 7.3 to find the least squares approximating line for bounce height b as a linear function of initial height h.Table 7.3
Hooke's Law states that the length L of a spring is a linear function of the force F applied to it. (See Figure 7.1 7 and Example 6.92.) Accordingly, there are constants a and b such thatL = a + bFTable 7.4 shows the results of attaching various weights to a spring.Table 7.17Table 7.4 (a) Determine
Table 7.5 gives life expectancies for people born in the United States in the given years.(a) Determine the least squares approximating line for these data and use it to predict the life expectancy of someone born in 2000.(b) How good is this model? Explain.Table 7.5
When an object is thrown straight up into the air, Newton's Second Law of Motion states that its height s (t) at time t is given byS(t) = s0 + v0t + 1/2gt2where v0 is its initial velocity and g is the constant of acceleration due to gravity. Suppose we take the measurements shown in Table 7.6.(a)
Table 7.7 gives the population of the United States at 10-year intervals for the years 1 950-2000. (a) Assuming an exponential growth model of the form p(t) = cekt, where p(t) is the population at time t, use least squares to find the equation for the growth rate of the population. (b) Use the
Table 7.8 shows average major league baseball salaries for the years 1970-2005. (a) Find the least squares approximating quadratic for these data. (b) Find the least squares approximating exponential for these data. (c) Which equation gives the better approximation? Why? (d) What do you estimate
A 200 mg sample of radioactive polonium-2 1 0 is observed as it decays. Table 7.9 shows the mass remaining at various times.Assuming an exponential decay model, use least squares to find the half-life of polonium-2 10. (See Section 6.7.)
Find the plane z = a + bx + cy that best fits the data points (0, - 4, 0), (5, 0, 0), (4, - 1 , 1), (1 , - 3, 1), and (- 1 , - 5, - 2).
In Exercises 1-3, find the standard matrix of the orthogonal projection onto the subspace W. Then use this matrix to find the orthogonal projection of v onto W.1.2. 3.
In Exercises 1-2, consider the data points (- 5, 3), (0, 3), (5, 2), and (1 0, 0). Compute the least squares error for the given line. In each case, plot the points and the line. 1. y = 3 - 1/3x 2. y = 5/2.
Verify that the standard matrix of the projection onto W in Example 7.31 (as constructed by Theorem 7.11) does not depend on the choice of basis. Takeas a basis for W and repeat the calculations to show that the resulting projection matrix is the same.
Let A be a matrix with linearly independent columns and let P = A (ATA)-1 AT be the matrix of orthogonal projection onto col(A). (a) Show that P is symmetric. (b) Show that P is idempotent.

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