New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
linear algebra

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

Use n + 1 equally spaced data points to interpolate f(t) = 1/(1 + t2) on an interval -a ≤ t ≤ a for a = 1, 1.5, 2. 2.5, 3, and n = 2, 4. 10. 20. Do all intervals exhibit the pathology illustrated in Figure 4.10? If not. how large can a be before the inter-polants have poor approximation
Repeat Exercise 4.4.24 for the hyperbolic secant function f(t) = sech t = 1/ cosh t.
(a) Give an example of an interpolating polynomial through n + 1 points that has degree < n.(b) Can you explain, without referring to the explicit formulae, why all the Lagrange interpolating polynomials based on n + 1 points must have degree equal to n?
(a) Prove that a polynomial p(x) = a0 + a1x + a2x2 + ∙∙∙∙∙ + anxn of degree < n vanishes at n-fl distinct points, so p(x1) = p(x2) =∙∙∙∙∙∙∙∙∙∙∙= p(xn+1) = 0, if and only if p(x) = 0 is the zero polynomial.(b) Prove that the monomials I,x,x2,... ,xn are linearly
Prove the determinant formulafor the (n + 1) x (n + 1) Vandermonde matrix defined in (4.43).
Let .x1.......xn be distinct real numbers. Prove that the n Ã— n matrix K with entries
Numerical Differentiation: The most common numerical methods for approximating the derivatives of a function are based on interpolation. To approximate the klh derivative f(k)(x0) at a point xo, one replaces the function f(x) by an interpolating polynomial pn(x) of degree n ≥ k based on the
are based on interpolation. One chooses ti -1- 1 interpolation points a
Given the data valuesconstruct the trigonometric function of the form g(t) = a cos Ï€ t + b sin Ï€ t that best approximates the data in the least squares sense.
(a) Find the exponential function of the form g(t) = ae' + be2' that best approximates t2 in the least squares sense based on the sample points 0, 1,2. 3.4.(b) What is the least squares error?(c) Compare the graphs on the interval [0,4]- where is the approximation the worst?(d) How much better can
(a) Find the best trigonometric approximation of the form g(t) = r cos(t + δ) to t2 using 5 and 9 equally spaced sample points on [0, π)} (b) Can you answer the question for g(t) = R1 cos(r + δ1) + r2 cos(2t + δ2)?
A trigonometric polynomial of degree n is a function of the formP(t) = a0 + a1cos t + b1 sin t + a2 cos 2t+ b2sin2t + ............+ an cos n t + bn sin n t.where a0,a1.b1........an,bn are the coefficients. Find the trigonometric polynomial of degree n that is the least squares approximation to the
The sine functions are defined aswhile Sj(x) = S0(x - j h) whenever h > 0 and j is an integer. We will interpolate a function /(x) at the mesh points Xj = jh, j = 0,... ,«, by a linear combination of sine functions: S(.v) = c0S0(x) cnSn (x). What are the coefficients Cj? Graph and discuss the
Re-solve Exercise 4.4.14 using the respective weight 2, 1. .5 at the three data points.
The amount of waste (in millions of tons a day) generated in Lower Slobbovia from 1960 to 1995 was(a) Find the equation for the least squares line that best fits these data. Use the result to estimate the amount of waste in the year 2000, and in the year 2005.(b) Redo your calculations using an
Find the best linear least squares fit of the following data using the indicated weights:(a)(b) (c) (d)
Write out an explicit formula for the weighted least squares error.
Approximate the function f(t) = 3√t using the least squares method based on the L2 norm on the interval [0, 1 ] by (a) A straight line (b) A parabola (c) A cubic polynomial
Approximate the function f(t) = 1/8(2f - 1 )3 + | by a quadratic polynomial on the interval [ - 1, 1 ] using the least squares method based on the L2 norm. Compare the graphs. Where is the error the largest?
For the function f(t) = sin f determine the best approximating linear and quadratic polynomials that minimize the least squares error based on the L2 norm on [ 0 '1/2π]
(a) Find the quadratic interpolant to f(x) = x5 on the interval [0, 1 ] based on equally spaced data points.(b) Find the quadratic least squares approximation based on the data points 0, .25, .5, .75, 1.(c) Find the quadratic least squares approximation with respect to the L2 norm.(d) Discuss the
Let g1(t).......gn(t) be prescribed, linearly independent functions. Explain how to best approximate a function f(t) by a linear combination Cngn (t) when the least squares error is measured in a weighted L2 norm ||f||2 = ʆbaf(t)2 w(t) dt with weight function w(t) > 0
A 20-pound turkey that has been thawed at the room temperature of 72° is placed in the oven at 1:00 pm. The temperature of the turkey is observed in 20 minute intervals to be 79°, 88°, and 96°. A turkey is cooked when its temperature reaches 165°. How much longer do you need to wait until the
(a) Find the quadratic least squares approximation to /(f) = f5 on the interval [0, 1 ] with weights(i) w(r) = 1 (ii) w(t) = t (iii) w(t) = e-t(b) Compare the errors-which gives the best result over the entire interval?
Let fa(x) =a.where ||ˆ™||2 denotes the L2 norm on (- ˆž,ˆž)(b) ||fa||ˆž = ˆša , where ||ˆ™|| denotes the Lˆž norm on (- ˆž, ˆž).(c) Use this example to explain why having a small least squares error does not necessarily mean that the functions are everywhere close.
Find the plane z = a + fix + yy that best approximates the following functions on the squareS = {-1 ‰¤ x ‰¤ 1, -1 ‰¤ y ‰¤ 1using the L2 normto measure the least squares error: (a) x2 + y2 (b) x3 - y3 (c) Sin Ï€ y sin Ï€ y
The amount of radium-224 in a sample was measured at the indicated times.(a) Estimate how much radium will be left after 10 days. (b) If the sample is considered to be safe when the amount of radium is less than .01 mg, estimate how long the sample needs to be stored before it can be safely
A sample of lead-210 measured the following radioactivity data at the given times:(a) Estimate the half-life of the lead-210.(b) How long until only 1 % of the original amount remains?
The following table gives the population of the United States for the years 1900-1950.(a) Use an exponential growth model to predict the population in 2000. 2010. and 2050.(b) The actual population for the year 2000 has recently been estimated to be 281 million. How does this affect your prediction
For the data points(a) determine the best plane z = a + bx + cy which fits the data in the least squares sense;(b) how would you answer the question in part (a) if the plane is constrained to go through the point x = 2, y = 2, z = 0?
Let R2 have the standard dot product. Classify the following pairs of vectors as(i) Basis,(ii) Orthogonal basis, and/or(iii) Orthonormal basis:(a)(b) (c) (d) (e) (f)
The cross product between vectors in R3 is defined by the formulaWhere (a) Show that u = v Ã— w is orthogonal, under the dot product, to both v and w. (b) Show that v Ã— w = 0 if and only if v and w are parallel. (c) Prove that if v, w ˆŠ R3 are orthogonal nonzero
Given angles θ,
(a) Show that v1,... , vn form an orthonormal basis of Rn for the inner product (v, w) = vT K w for K > 0 if and only if AT KA = I where A = (v1, v2 . . . vn). (b) Prove that any basis of Rn is an orthonormal basis with respect to some inner product. Is the inner product uniquely determined? (c)
Describe all orthonormal bases of R2 for the inner products(a)(b)
Let v and w be elements of an inner product space. Prove that ||v + w||2 = ||v||2 + ||w||2 if and only if v, w are orthogonal. Explain why this formula can be viewed as the generalization of the Pythagorean Theorem.
Prove that if v1, v2 form a basis of an inner product space V and ||v1|| = ||v2|| then v1 + v2 and v1 - v2 form an orthogonal basis of V.
Suppose v1, . . . ,vk are nonzero mutually orthogonal elements of an inner product space V. Write down their Gram matrix. Why is it nonsingular?
Let V - P(1) be the vector space consisting of linear polynomials p(t) = at + b.(a) Carefully explain whydefines an inner product on V. (b) Find all polynomials p(t) = a t + b ˆŠ V that are orthogonal to p1(t) = 1 based on this inner product. (c) Use part (b) to construct an orthonormal
Explain why the functions cos x, sin x form an orthogonal basis for the space of solutions to the differential equation y" + y = 0 under the L2 inner product on [- π, π].
Do the functions ex/2, e-x/2 form an orthogonal basis for the space of solutions to the differential equation 4y" - y = 0 under the L2 inner product on [0, 1]? If not, can you find an orthogonal basis of the solution space?
(a) Prove thatform an orthonormal basis for R3 for the usual dot product. (b) Find the coordinates of v = (1, 1, 1)T relative to this basis. (c) Verify formula (5.5) in this particular case.
(a) Prove that the vectorsform an orthogonal basis of R3 with the dot product. (b) Use orthogonality to write the vector v = (1, 2, 3)T as a linear combination of v1, v2, v3. (c) Verify the formula (5.8) for ||v||. (d) Construct an orthonormal basis, using the given vectors. (e) Write v as a linear
Let R2 have the inner product defined by the positive definite matrix(a) Show that v1 = (1, 1)T, v2 = (- 2, 1)T form an orthogonal basis. (b) Write the vector v = (3, 2)T as a linear combination of v1, v2 using the orthogonality formula (5.7). (c) Verify the formula (5.8) for ||v||. (d) Find an
Find an example that demonstrates why equation (5.5) is not valid for a non-orthonormal basis.
Use orthogonality to write the polynomials 1, x and x2 as linear combinations of the orthogonal basis (5.1).
(a) Prove that the polynomialsform an orthogonal basis for the vector space P3 of cubic polynomials for the L2 inner product (b) Find an orthonormal basis of V3. (c) Write t3 as a linear combination of P0, P1, P2, P3 using the orthogonal basis formula (5.7).
(a) Prove that the polynomialsform an orthogonal basis for P2 with respect to the weighted inner product (b) Find the corresponding orthonormal basis. (c) Write t2 as a linear combination of P0, P1, P2 using the orthogonal basis formula (5.7).
Write the following trigonometric polynomials in terms of the basis functions (5.11): (a) cos2 a: (b) cos x sin x (c) sin3x (d) cos2 sin3x (e) cos4x
Repeat Exercise 5.1.1, but use the weighted inner product (v, w) = v1w1 + 1/9 v2 w2 instead of the dot product.Exercise 5.1.1.Let R2 have the standard dot product. Classify the following pairs of vectors as(i) Basis,(ii) Orthogonal basis, and/or(iii) Orthonormal basis:(a)(b) (c) (d) (e) (f)
Show that the 2n + 1 complex exponentials eikx for k = - n, - n + 1........ - 1, 0, 1,... ,n, form an orthonormal basis for the space of complex-valued trigonometric polynomials under the Hermitian inner product
Prove the trigonometric integral identities (5.13).
Fill in the complete details of the proof of Theorem 5.9.If v1,... , vn form an orthogonal basis, then the corresponding coordinates of a vectorIn this case, its norm can be computed using the formula Equation (5.7), along with its orthonormal simplification (5.4), is one of the most useful
Show that the standard basis vectors e1, e2, e3 form an orthogonal basis with respect to the weighted inner product (v, w) = v1 w1 + 2 u2 w2 + 3 v3 w3 on R3. Find an orthonormal basis for this inner product space.
Find all values of a so that the vectorsform an orthogonal basis of R2 under (a) The dot product; (b) The weighted inner product (v, w) = 3v1w1 + 2v2w2 (c) The inner product prescribed by the positive definite matrix
True or false: If v1, v2, v3 are a basis for M3, then they form an orthogonal basis under some appropriately weighted inner product (v, w) = a v1w1 + b v2w2 + c v3w3.
Use the Gram-Schmidt process to determine an orthonormal basis for R3 starting with the following sets of vectors:(a)(b) (c)
Construct an orthonormal basis for R3 with respect to the inner products defined by the following positive definite matrices:(a)(b)
Verify that the Gram-Schmidt formulae (5.19) also produce an orthogonal basis of a complex vector space under a Hermitian inner product.
(a) Apply the complex Gram-Schmidt algorithm from Exercise 5.2.12 to produce an orthonormal basis starting with the vectors (1+ i. 1 - i)T, (1 - 2i,5i)T ∊ C2. (b) Do the same for (1 + i, 1 - i, 2 - i)T, (1 + 2i, - 2i, 2 - i)T, (1, 1 - 2i, i)T ∊ C3.
Use the complex Gram-Schmidt algorithm from Exercise 5,2,12 to construct orthonormal bases for (a) The subspace spanned by (1 - i, 1, 0)T, (0.3 - i, 2i)T; (b) The set of solutions to (2 - i)x - 2i y + (l - 2i)z = 0; (c) The subspace spanned by (- i, 1, - 1, i)T, (0, 2i, 1 - i, - 1 + i)T, (1, i, -
Suppose that V C Rn is a proper subspace, and u1........un, forms an orthonormal basis of V. Prove that there exist vectors um+1,... , un ∊ R" \ V such that the complete collection u1,... ,un forms an orthonormal basis for Rn.
Use the modified Gram-Schmidt process (5.26- 27) to produce orthonormal bases for the spaces spanned by the following vectors:(a)(b) (c) (d)
Use the Gram-Schmidt process to construct an orthonormal basis for R4 starting with the following sets of vectors: (a) (1, 0, 1,0)T, (0,1,0, - 1)T, (1,0,0,1)T, (1, 1,1, l)T (b) (1,0,0, l)T, (4, 1,0, 0)T, (1,0,2, l)T, (0,2,0, l)T,
Prove that (5.28) does indeed produce an orthonormal basis. Explain why the result is the same orthonormal basis as the ordinary Gram-Schmidt method.
Let wj(j) be the vectors in the stable Gram-Schmidt algorithm (5.28). Prove that the coefficients in (5.23) are given by rji = ||wi(i)||, and rij = (wj(i), ui) for i < j.
Try the Gram-Schmidt procedure on the vectorsWhat happens? Can you explain why you are unable to complete the algorithm?
Use the Gram-Schmidt process to construct an orthonormal basis for the following subspaces of R3: (a) The plane spanned by (0. 2, l)T, (1, - 2, - l)T; (b) The plane defined by the equation 2x - y + 3z = 0: (c) The set of all vectors orthogonal to (1, - 1, - 2)T.
Find an orthonormal basis for the following subspaces of R4:(a) The span of the vectors(b) The kernel of the matrix (c) The corange of the preceding matrix (d) The range of the matrix (e) The cokernel of the preceding matrix (f) The set of all vectors orthogonal to (1, 1,-1,-1)T
Find orthonormal bases for the four fundamental subspaces associated with the following matrices:(a)(b) (c) (d)
Redo Exercise 5.2.1 using(i) The weighted inner product (v, w) = 3 v1 w1 + 2 v2 w2 + v3 w3:(ii) The inner product induced by the positive definite matrix
Construct an orthonormal basis of R2 for the nonstandard inner products(a)(b) (c)
Determine which of the following matrices are(i) Orthogonal;(ii) Proper orthogonal.(a)(b) (c) (d) (e) (f) (g)
True or false: (a) A matrix whose columns form an orthogonal basis of Rn is an orthogonal matrix. (b) A matrix whose rows form an orthonormal basis of Rn is an orthogonal matrix. (c) An orthogonal matrix is symmetric if and only if it is a diagonal matrix.
Prove that an upper triangular matrix U is orthogonal if and only if U is a diagonal matrix. What are its diagonal entries?
(a) Show that the elementary row operation matrix corresponding to the interchange of two rows is an improper orthogonal matrix. (b) Are there any other orthogonal elementary matrices?
(a) Prove that every permutation matrix is orthogonal. (b) How many permutation matrices of a given size are proper orthogonal?
(a) Prove that if Q is an orthogonal matrix, then ||Qx|| = ||x|| for any vector x ∊ Rn, where ||∙|| denotes the standard Euclidean norm. (b) Prove the converse: if ||(Qx|| = ||x|| for all x ∊ Rn, then Q is an orthogonal matrix.
(a) Show that if u ∊ Rn is a unit vector, then the n × n matrix Q = 1 - 2uuT is an orthogonal matrix, known as an elementary reflection or Householder matrix. (b) Write down the elementary reflection matrices corresponding to the following unit vectors: (i) (1, 0)T (ii) (3/5, 4/5)T (iii) (0, 1,
Show that if AT = - A is any skew-symmetric matrix, then its Cayley Transform Q = (I - A)-1 (I + A) is an orthogonal matrix. Can you prove that I - A is always invertible?
Suppose S is an n x n matrix whose columns form an orthogonal, but not orthonormal, basis of Rn. (a) Find a formula for S-1 mimicking the formula Q-1 = QT for an orthogonal matrix. (b) Use your formula to determine the inverse of the wavelet matrix W whose columns form the orthogonal wavelet basis
(a) Show thata reflection matrix, and representing a rotation by angle Î¸ around the z axis, are both orthogonal. (b) Verify that the products R Q and QR are also orthogonal. (c) Which of the preceding matrices, R, Q, R Q, Q R are proper orthogonal?
Let v1,... , vn and w1.......wn be two sets of linearly independent vectors in Rn. Show that all their dot products are the same, so vi ∙ vj = wi ∙ wj for all 1, 7 = 1........n, if and only if there is an orthogonal matrix Q such that w, = Q vi for all i = 1,... , n.
Suppose u1,.......,uk form an orthonormal set of vectors in Rn with k < n. Let Q = (u, u2 ... uk) denote the n × k matrix whose columns are the orthonormal vectors. (a) Prove that QT Q = Ik. (b) Is QQT = In?
Let A be an m x n matrix whose columns are nonzero, mutually orthogonal vectors in Rm. (a) Explain why m > n. (b) Prove that AT A is a diagonal matrix. What are the diagonal entries? (c) Is A AT diagonal?
Let K > 0 be a positive definite n x n matrix. Prove that an n × n matrix S satisfies ST K S = I if and only if the columns of S form an orthonormal basis of Rn with respect to the inner product (v, w) = vT K w.
A set of n × n matrices G ⊂ Mn×n, is said to form a group if(1) Whenever A, B ∊ G, so is the product A B ∊ G. and(2) Whenever A ∊ G. then A is nonsingular, and A-1 ∊ G.(a) Show that I ∊ G.(b) Prove that the following sets of n × n matrices form a group:(i) All nonsingular
(a) Show that U is a unitary matrix if and only if U-1 = U€ .(b) Show that the following matrices are unitary and compute their inverses:(i)
Write down the QR matrix factorization corresponding to the vectors in Example 5.17.
Find the Q R factorization of the following matrices:(a)(b) (c) (d) (e) (f)
For each of the following linear systems:(a) Find the Q R factorization of the coefficient matrix, and then(b) Use your factorization to solve the system.(i)(ii) (iii)
Use the numerically stable version of the Gram-Schmidt process to find the QR factorizations of the 3 × 3, 4 × 4 and 5 × 5 versions of the tridiagonal matrix that has 4's along the diagonal and l's on the sub- and super-diagonals, as in Example 1.37.
True or false: (a) If Q is an improper 2 × 2 orthogonal matrix, then Q2 = I. (b) If Q is an improper 3 × 3 orthogonal matrix, then Q2 = I.
Use Householder's Method to solve Exercises 5.3.27 and 5.3.29.
(a) How many arithmetic operations are required to compute the Q R factorization of an n × n matrix?(b) How many additional operations are needed to utilize the factorization to solve a linear system Ax = b via (5.34)?(c) Compare the amount of computational effort with standard Gaussian
Prove that the Q R factorization of a matrix is unique if all the diagonal entries of R are assumed to be positive.
(a) Show that applying the Gram-Schmidt algorithm to the columns of A produces an orthonormal basis for mg A.(b) Prove that this is equivalent to the matrix factorization A = Q R, where Q is an m Ã— n matrix with orthonormal columns, while R is a nonsingular n x n upper triangular matrix.(c)
(a) According to Exercise 5.2.12, the Grarn-Schmidt process can also be applied to produce orthonormal bases of complex vector spaces. In the case of Cn, explain how this is equivalent to the factorization of a nonsingular complex matrix A = U R into the product of a unitary matrix U (see Exercise
Write out a pseudocode program to implement Householder's Method. The input should be an n × n matrix A and the output should be the Householder unit vectors u1........un-1 and the upper triangular matrix R. Test your code on one of the examples in Exercises 5.3.26-28.
(a) Prove that, for any θ,

Showing 8300 - 8400 of 11883

First
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved