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Linear Algebra A Modern Introduction 3rd edition David Poole - Solutions

Find an SVD of the indicated matrix.A in Exercise 3Data from Exercise 3 A =
Find the least squares approximating line for the given points and compute the corresponding least squares error.(-5, -1), (0, 1), (5, 2), (10, 4)
Show that the matrix is ill-conditioned. 1 0.1 0.11 A = 0.1 0.11 0.111 0.111 0.1111 0.11
Find an SVD of the indicated matrix. -2 0
Find the best linear approximation to f on the interval [0, 1]. f(x) = sin(πx/2)
Find the least squares approximating line for the given points and compute the corresponding least squares error.(-5, 3), (0, 3), (5, 2), (10, 0)
Find the least squares approximating line for the given points and compute the corresponding least squares error.(1, 1), (2, 3), (3, 4), (4, 5), (5, 7)
Find an SVD of the indicated matrix. A =
Find the least squares approximating line for the given points and compute the corresponding least squares error.(1, 10), (2, 8), (3, 5), (4, 3), (5, 0)
Find the least squares solution of 2 X1 -1 -1 X2 _0 3
Find an SVD of the indicated matrix.A in Exercise 5Data From Exercise 5 3 A = [4
Find the orthogonal projection ofonto the column space of 3 A = | 0 1
Find an SVD of the indicated matrix.A in Exercise 6Data From Exercise 6A = [3 4]
Find the best quadratic approximation to f on the interval [0,1].f(x) = sin(πx/2)
Find the least squares approximating parabola for the given points.(1, 8), (2, 7), (3, 5), (4, 2)
Determine which of the four inner product axioms do not hold. Give a specific example in each case.In P2, define {p(x), q(x)} = p(0) q(0).
Find an SVD of the indicated matrix.A in Exercise 7Data From Exercise 7 3 -2 0
Find the least squares approximating parabola for the given points.(-2, 4), (-1, 7), (0, 3), (1, 0), (2, -1)
Determine which of the four inner product axioms do not hold. Give a specific example in each case.In P2, define {p(x), q(x)} = p(1) q(1).
Prove that || || defines a norm on the vectors space V. V = C[0, 1], ||f|| = F(x)|dx
Prove that ∥ ∥ defines a norm on the vectors space V. max |f(x)| 0
Find an SVD of the indicated matrix.A in Exercise 8Data From Exercise 8 A = -2 2
Find the least squares approximating parabola for the given points.(-2, 0), (-1, -11 ), (0, -10), (1, -9), (2, 8)
Find (a) the singular values, (b) a singular value decomposition, and (c) the pseudo inverse of the matrix A. 1 |A = -1 -1
Determine which of the four inner product axioms do not hold. Give a specific example in each case.In M22, define {A,B} = det(AB).
Prove Theorem 7.5(b).
Find an SVD of the indicated matrix.A in Exercise 9Data From Exercise 9 |2 0 A = 2 0
Compute ||A||F, ||A||1, and ||A||ˆž. 2 3 4 1
Find an SVD of the indicated matrix. A =
Show that ||1||2 = 2π and ||cos kx||2 = π in e[- π,π].
Compute ||A||F, ||A||1, and ||A||ˆž. -1 A = 3
Find the outer product form of the SVD for the matrix in the given exercises.Exercises 3 and 11Data From Exercise 3 and 11 A 0.
Find a least squares solution of Ax = b by constructing and solving the normal equations. -2 -3 |,b = A = 2 -2 3 4
Compute ||A||F, ||A||1, and ||A||ˆž. 5 A = -2 %3D -1
Find the outer product form of the SVD for the matrix in the given exercises.Exercise 14Data From Exercise 14 A = I|
Find a least squares solution of Ax = b by constructing and solving the normal equations. -1 ,b = A = 3 -1 3_ 2 -1 2.
Compute ||A||F, ||A||1, and ||A||ˆž. 2 1 1 3 2 A = [1 1 3,
Find the outer product form of the SVD for the matrix in the given exercises.Exercise 7 and 17Data From Exercise 7 and 17 -2 0
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 -3 ,b = 2 |A = 0 -1 -1 4
Compute ||A||F, ||A||1, and ||A||ˆž. 0 -5 2 1 -3 3 -4 -4 3
Find the outer product form of the SVD for the matrix in the given exercises.Exercises 9 and 19Data From Exercise 9 and 19 2 0 A : 2 0
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 -1 3 ,b = -1
Compute ||A||F, ||A||1, and ||A||ˆž. 4 -2 -1 A = | 0 -1 -3 0_
Find the best approximation to a solution of the given system of equations. x + y - z = 2 -y + 2z = 6 3x + 2y - z = 11 -x +
Find vectors x and y with ||x||s= 1 and ||y||m= 1 such that ||A||1= ||Ax||sand ||A||ˆž= ||Ay||m, where A is the matrix in the given exercise.Exercise 20Data From Exercise 20 2 3 A =
Find the Fourier coefficients a0, ak, and bk of f on [-π, π].f(x) = |x|
Find the best approximation to a solution of the given system of equations. 2x + 3y + z = 21 x + y + z = 7 -x + y - z =
Find vectors x and y with ||x||s= 1 and ||y||m= 1 such that ||A||1= ||Ax||sand ||A||ˆž= ||Ay||m, where A is the matrix in the given exercise.Exercise 21Data From Exercise 21 -1 A = – 3 3
Find vectors x and y with ||x||s= 1 and ||y||m= 1 such that ||A||1= ||Ax||sand ||A||ˆž= ||Ay||m, where A is the matrix in the given exercise.Exercise 22Data From Exercise 22 A = -2 -1
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|| = 2Evaluate the expressions||2u - 3v + w||
Find vectors x and y with ||x||s= 1 and ||y||m= 1 such that ||A||1= ||Ax||sand ||A||ˆž= ||Ay||m, where A is the matrix in the given exercise.Exercise 23Data From Exercise 23 A = 3
Find the minimum distance of the codes. C =
Find vectors x and y with ||x||s= 1 and ||y||m= 1 such that ||A||1= ||Ax||sand ||A||ˆž= ||Ay||m, where A is the matrix in the given exercise.Exercise 24Data From Exercise 24 -5 2 1 -3 3 %3D -4 -4 3
Find the minimum distance of the codes. BIBIGNAR 1. _0
Find vectors x and y with ||x||s= 1 and ||y||m= 1 such that ||A||1= ||Ax||sand ||A||ˆž= ||Ay||m, where A is the matrix in the given exercise.Exercise 25Data From Exercise 25 4 -2 -1 A = | 0 -1 [3 -3
Find the minimum distance of the codes.The even parity code En.
Find the minimum distance of the codes.The n-times repetition code Repn.
Find the minimum distance of the codes.The code with parity check matrix P = [I | A], where 1 0 1 1 0 1 0 1 1 1 1 1 1 10 0 Lo
In exercises (u,v) is an inner product. prove that the given statement is an identity {u, v} = 1/4 ||u + v||2 - 1/4|| u - v||2
Let ||A|| be a matrix norm that is compatible with a vector norm ||x||. Prove that ||A|| ≥ |λ| for every eigenvalue λ of A.
Find the minimum distance of the codes.The code with parity check matrix 1 0 0 P = | 1 1 1 1 1 0 0 1
In exercises (u,v) is an inner product. prove that the given statement is an identity Prove that ||u + v || = ||u - v|| if and only if u and v are orthogonal.
Compute the minimum distance of the code C and decode the vectors u, v, and w using nearest neighbor decoding. C = u =|0 |, v =| 0 w = | 0
(u,v) is an inner product. prove that the given statement is an identity Prove that d(u, v) = √||u||2 + ||v||2 if and only if u and v are orthogonal.
Compute the minimum distance of the code C and decode the vectors u, v, and w using nearest neighbor decoding.C has generator matrix G = 0 |, u = | 0 v = | 0
Compute (a) ||A||2 and (b) cond2(A) for the indicated matrix.A in Exercise 3Data From Exercise 3 A =
Construct a linear (n, k, d) code or prove that no such code exists.n = 8, k = 1, d = 8
Compute (a) ||A||2 and (b) cond2(A) for the indicated matrix.A in Exercise 8Data From Exercise 8 A = -2 2
Construct a linear (n, k, d) code or prove that no such code exists.n = 8, k = 2, d = 8
Find cond1(A) and condˆž (A). State whether the given matrix is ill-conditioned. 150 200 A = 3001 4002
Compute (a) ||A||2 and (b) cond2(A) for the indicated matrix. 0.9
Construct a linear (n, k, d) code or prove that no such code exists.n = 8, k = 5, d = 5
Find cond1(A) and condˆž (A). State whether the given matrix is ill-conditioned. A = | 5 6 5
Apply the Gram-Schmidt Process to the basis B to obtain an orthogonal basis for the inner product space V relative to the given inner product.V = P2 [0,1], B = {1, 1 + x, 1 + x + x2}, with the inner product in Example 7.5.
Compute (a) ||A||2 and (b) cond2(A) for the indicated matrix. 10 10 0 A = 100 100
Construct a linear (n, k, d) code or prove that no such code exists.n = 8, k = 4, d = 4
Find cond1(A) and condˆž (A). State whether the given matrix is ill-conditioned. A = ||
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. W = span , v = | 0 L0]
Compute the pseudo inverse A+of A in the given exercise.Exercise 3Data From Exercise 3 1 A
Let be defined by T : M22†’ R be defined by T(A) = tr (A).(a) Which, if any, of the following matrices are in ker(T)?(i)(ii)(iii)(b) Which, if any, of the following scalars are in range(T)?(i) 0 (ii) 5(iii) -ˆš2(c) Describe ker(T) and range(T). 3 -2 2 1
(a) Find the coordinate vectors [x]Band [x]Bof x with respect to the bases B and C, respectively.(b) Find the change-of-basis matrix PC†Bfrom B to C.(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).(d) Find the change-of-basis matrix
Test the sets of matrices for linear independence in M22. For those that are linearly dependent, express one of the matrices as a linear combination of the others. -1 -2 1]' | 2 4 2 -3 2 3 3
(a) Find the coordinate vectors [x]Band [x]Bof x with respect to the bases B and C, respectively.(b) Find the change-of-basis matrix PC†Bfrom B to C.(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).(d) Find the change-of-basis matrix
(a) Find the coordinate vectors [x]Band [x]Bof x with respect to the bases B and C, respectively.(b) Find the change-of-basis matrix PC†Bfrom B to C.(c) Use your answer to part (b) to compute and compare your answer with the one found in part (a).(d) Find the change-of-basis matrix
Test the sets of matrices for linear independence in M22. For those that are linearly dependent, express one of the matrices as a linear combination of the others. -1 -1
Find the solution of the differential equation that satisfies the given boundary condition(s).x'' + x' - 12x = 0, x(0) = 0, x'(0) = 1
Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, list all of the axioms that fail to hold.R2, with the usual addition but scalar multiplication defined by х сх - У
Follow the instructions for Exercises 1–4 using p(x) instead of x.p(x) = 3 + 2x, B = {1 + x, 1 - x}, C = {2x, 3} in P1 Instructions From Exercise 1(a) Find the coordinate vectors [x]B and [x]B of x with respect to the bases B and C, respectively.(b) Find the
Find the matrix [T]C†Bof the linear transformation T : V †’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem. T : P2 †’ R2 defined by T(P(x) ) B = {x2, x,
Find the solution of the differential equation that satisfies the given boundary condition(s).g'' - 2g = 0, g(0) = 1, g(1) = 0
Follow the instructions for Exercises 1–4 using p(x) instead of x.p(x) = 1 + x2, B = {1 + x + x2, x + x2, x2}, C = {1, x, x2} in P2 Instructions From Exercise 1(a) Find the coordinate vectors [x]B and [x]B of x with respect to the bases B and C, respectively.(b) Find the change-of-basis
Find the matrix [T]C†Bof the linear transformation T : V †’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem. T : M22 †’ M22 defined by T(A) = AT, B = C =
Find the solution of the differential equation that satisfies the given boundary condition(s).y'' - 2ky' + k2y = 0, k ≠ 0, y (0) = 1, y (1) = 0
Find the matrix [T]C†Bof the linear transformation T : V †’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem. Repeat Exercise 9 with B = {E22,E21,E12,E11,}, and C
Find the solution of the differential equation that satisfies the given boundary condition(s).f '' - 2f ' + 5f = 0, f (0) = 1, f (π/4) = 0
Determine whether T is a linear transformation. T : F → F defined by T(f) = f(x))2
Find the matrix [T]C†Bof the linear transformation T : V †’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem. T : M22 †’ M22 defined by T(A) = AB - BA, where B
Find the solution of the differential equation that satisfies the given boundary condition(s).h'' - 4h' + 5h = 0, h(0) 0, h'(0) = -1
Find the matrix [T]C†Bof the linear transformation T : V †’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v) directly and using the theorem. T : M22 †’ M22 defined by T(A) = A - AT, B = C =
Test the sets of functions for linear independence in F. For those that are linearly dependent, express one of the functions as a linear combination of the others.{1, ln(2x), ln(x2)}

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