EBK LINEAR ALGEBRA AND ITS APPLICATIONS
6th Edition
ISBN: 9780135851043
Author: Lay
Publisher: PEARSON CO
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Chapter 6.5, Problem 21E
To determine
To find: The statement is True or False.
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Consider a forest where the population of a particular plant species grows exponentially. In a real-world
scenario, we often deal with systems where the analytical function describing the phenomenon is not available.
In such cases, numerical methods come in handy.
For the sake of this task, however, you are provided with an analytical function so that you can compare
the results of the numerical methods to some ground truth. The population P(t) of the plants at time t (in
years) is given by the equation:
P(t) = 200
0.03 t
You are tasked with estimating the rate of change of the plant population at t = 5 years using numerical
differentiation methods. First, compute the value of P'(t) at t = 5 analytically. Then, estimate P'(t) at t = 5
years using the following numerical differentiation methods:
⚫ forward difference method (2nd-order accurate)
3
⚫ backward difference method (2nd-order accurate)
⚫ central difference method (2nd-order accurate)
Use h = 0.5 as the step size and round all…
Nicole organized a new corporation. The corporation began business on April 1 of year 1. She made the following
expenditures associated with getting the corporation started:
Expense
Date
Amount
Attorney fees for articles of incorporation
February 10 $ 40,500
March 1-March 30 wages
March 30
6,550
March 1-March 30 rent
Stock issuance costs
March 30
2,850
April 1-May 30 wages
Note: Leave no answer blank. Enter zero if applicable.
April 1
May 30
24,000
16,375
c. What amount can the corporation deduct as amortization expense for the organizational expenditures and for the start-up costs for
year 1 [not including the amount determined in part (b)]?
Note: Round intermediate calculations to 2 decimal places and final answer to the nearest whole dollar amount.
Start-up costs amortized
Organizational expenditures amortized
Last Chance Mine (LCM) purchased a coal deposit for $2,918,300. It estimated it would extract 18,950 tons of coal from the
deposit. LCM mined the coal and sold it, reporting gross receipts of $1.24 million, $13 million, and $11 million for years 1
through 3, respectively. During years 1-3, LCM reported net income (loss) from the coal deposit activity in the amount of
($11,400), $550,000, and $502,500, respectively. In years 1-3, LCM extracted 19,950 tons of coal as follows:
(1) Tons of
Coal
18,950
Depletion
(2) Basis (2)(1) Rate
$2,918,300 $154.00
Tons Extracted per Year
Year 1
4,500
Year 2
8,850
Year 3
6,600
Note: Leave no answer blank. Enter zero if applicable. Enter your answers in dollars and not in millions of dollars.
a. What is LCM's cost depletion for years 1, 2, and 3?
Cost Depletion
Year 1
Year 2
Year 3
Chapter 6 Solutions
EBK LINEAR ALGEBRA AND ITS APPLICATIONS
Ch. 6.1 - Let a = [21] and b = [31]. Compute abaa and...Ch. 6.1 - Let c = [4/312/3] and d = [561]. a. Find a unit...Ch. 6.1 - Prob. 3PPCh. 6.1 - Compute the quantities in Exercises 1—8 using...Ch. 6.1 - Prob. 2ECh. 6.1 - Compute the quantities in Exercises 1—8 using...Ch. 6.1 - Compute the quantities in Exercises 1—8 using...Ch. 6.1 - Compute the quantities in Exercises 1—8 using...Ch. 6.1 - Compute the quantities in Exercises 1—8 using...Ch. 6.1 - Compute the quantities in Exercises 1—8 using...
Ch. 6.1 - Prob. 8ECh. 6.1 - In Exercises 912, find a unit vector in the...Ch. 6.1 - Prob. 10ECh. 6.1 - In Exercises 912, find a unit vector in the...Ch. 6.1 - Prob. 12ECh. 6.1 - Find the distance between x = [103] and y = [15].Ch. 6.1 - Prob. 14ECh. 6.1 - Determine which pairs of vectors in Exercises 1518...Ch. 6.1 - Determine which pairs of vectors in Exercises 1518...Ch. 6.1 - Determine which pairs of vectors in Exercises 1518...Ch. 6.1 - Determine which pairs of vectors in Exercises 1518...Ch. 6.1 - In Exercises 19—28, all vectors are in Rn. Mark...Ch. 6.1 - In Exercises 19—28, all vectors are in Rn. Mark...Ch. 6.1 - In Exercises 19—28, all vectors are in Rn. Mark...Ch. 6.1 - In Exercises 19—28, all vectors are in Rn. Mark...Ch. 6.1 - Prob. 23ECh. 6.1 - Prob. 24ECh. 6.1 - Prob. 25ECh. 6.1 - Prob. 26ECh. 6.1 - Prob. 27ECh. 6.1 - Use the transpose definition of the inner product...Ch. 6.1 - Prob. 30ECh. 6.1 - Let u = [251] and v = [746]. Compute and compare...Ch. 6.1 - Verify the parallelogram law for vectors u and v...Ch. 6.1 - Let v = [ab] Describe the set H of vectors [xy]...Ch. 6.1 - Let u = [567], and let W be the set of all x in 3...Ch. 6.1 - Suppose a vector y is orthogonal to vectors u and...Ch. 6.1 - Suppose y is orthogonal to u and v. Show that y is...Ch. 6.1 - Let W = Span {v1,,vp}. Show that if x is...Ch. 6.1 - Let W be a subspace of n, and let W be the set of...Ch. 6.1 - Show that if x is in both W and W, then x = 0.Ch. 6.2 - Let u1= [1/52/5] and u2= [2/51/5]. Show that {u1....Ch. 6.2 - Let y and L be as in Example 3 and Figure 3....Ch. 6.2 - Let U and x be as in Example 6. and let y = [326]....Ch. 6.2 - Let U be an n n matrix with orthonormal columns....Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - Prob. 4ECh. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 16, determine which sets of vectors...Ch. 6.2 - In Exercises 710, show that {u1, u2} or {u1, u2,...Ch. 6.2 - Prob. 8ECh. 6.2 - In Exercises 710, show that {u1, u2} or {u1, u2,...Ch. 6.2 - In Exercises 710, show that {u1, u2} or {u1, u2,...Ch. 6.2 - Compute the orthogonal projection of [17] onto the...Ch. 6.2 - Prob. 12ECh. 6.2 - Let y = [23] and u = [47] Write y as the sum of...Ch. 6.2 - Let y=26 and u=61. Write y as the sum of a vector...Ch. 6.2 - Let y = [31] and u = [86] Compute the distance...Ch. 6.2 - Let y = [39] and u = [12] Compute the distance...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 1722, determine which sets of vectors...Ch. 6.2 - In Exercises 23—32, all vectors are in Rn. Mark...Ch. 6.2 - Prob. 24ECh. 6.2 - Prob. 25ECh. 6.2 - Prob. 26ECh. 6.2 - Prob. 27ECh. 6.2 - Prob. 28ECh. 6.2 - Prob. 29ECh. 6.2 - Prob. 30ECh. 6.2 - Prob. 31ECh. 6.2 - Prob. 32ECh. 6.2 - Prove Theorem 7. [Hint: For (a), compute |Ux||2,...Ch. 6.2 - Suppose W is a sub space of n spanned by n nonzero...Ch. 6.2 - Let U be a square matrix with orthonormal columns....Ch. 6.2 - Let U be an n n orthogonal matrix. Show that the...Ch. 6.2 - Let U and V be n n orthogonal matrices. Explain...Ch. 6.2 - Let U be an orthogonal matrix, and construct V by...Ch. 6.2 - Show that the orthogonal projection of a vector y...Ch. 6.2 - Let {v1, v2} be an orthogonal set of nonzero...Ch. 6.2 - Prob. 41ECh. 6.2 - Given u 0 in n, let L = Span{u}. For y in n, the...Ch. 6.2 - Prob. 43ECh. 6.2 - In parts (a)—(d), let U be the matrix formed by...Ch. 6.3 - Let u1 = [714], u2 = [112], x = [916], and W =...Ch. 6.3 - Let W be a subspace of n. Let x and y be vectors...Ch. 6.3 - In Exercises 1 and 2, you may assume that {u1,,...Ch. 6.3 - u1 = [1211], u2 = [2111], u3 = [1121], u4 =...Ch. 6.3 - In Exercises 36, verify that {u1, u2} is an...Ch. 6.3 - In Exercises 3—6, verify that u1,u2 is an...Ch. 6.3 - In Exercises 36, verify that {u1, u2} is an...Ch. 6.3 - Prob. 6ECh. 6.3 - In Exercises 710, let W be the subspace spanned by...Ch. 6.3 - In Exercises 710, let W be the subspace spanned by...Ch. 6.3 - In Exercises 710, let W be the subspace spanned by...Ch. 6.3 - In Exercises 710, let W be the subspace spanned by...Ch. 6.3 - In Exercises 11 and 12, find the closest point to...Ch. 6.3 - In Exercises 11 and 12, find the closest point to...Ch. 6.3 - In Exercises 13 and 14, find the best...Ch. 6.3 - In Exercises 13 and 14, find the best...Ch. 6.3 - Let y = [595], u1 = [351], u2 = [321]. Find die...Ch. 6.3 - Let y, v1, and v2 be as in Exercise 12. Find the...Ch. 6.3 - Let y = [481], u1 = [2/31/32/3], u2 = [2/32/31/3],...Ch. 6.3 - Let y = [79], u1 = [1/103/10], and W = Span {u1}....Ch. 6.3 - Let u1 = [112], u2 = [512], and u3 = [001].Note...Ch. 6.3 - Let u1 and u2 be as in Exercise 19, and let u4 =...Ch. 6.3 - Prob. 21ECh. 6.3 - Prob. 22ECh. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - In Exercises 23—30, all vectors and subspaces...Ch. 6.3 - Prob. 26ECh. 6.3 - Prob. 27ECh. 6.3 - In Exercises 23—30, all vectors and subspaces...Ch. 6.3 - Prob. 29ECh. 6.3 - Prob. 30ECh. 6.3 - Let A be an mn matrix. Prove that every vector x...Ch. 6.3 - Prob. 32ECh. 6.3 - Prob. 35ECh. 6.3 - Prob. 36ECh. 6.3 - Prob. 37ECh. 6.3 - Prob. 38ECh. 6.4 - Let W = Span {x1, x2}, where x1 = [111] and x2 =...Ch. 6.4 - Suppose A = QR, where Q is an m n matrix with...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - In Exercises 1-6, the given set is a basis for a...Ch. 6.4 - Find an orthonormal basis of the subspace spanned...Ch. 6.4 - Find an orthonormal basis of the subspace spanned...Ch. 6.4 - Find an orthogonal basis for the column space of...Ch. 6.4 - Find an orthogonal basis for the column space of...Ch. 6.4 - Find an orthogonal basis for the column space of...Ch. 6.4 - Find an orthogonal basis for the column space of...Ch. 6.4 - In Exercises 13 and 14, the columns of Q were...Ch. 6.4 - In Exercises 13 and 14, the columns of Q were...Ch. 6.4 - Find a QR factorization of the matrix in Exercise...Ch. 6.4 - Find a QR factorization of the matrix in Exercise...Ch. 6.4 - Prob. 17ECh. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - Prob. 20ECh. 6.4 - Prob. 21ECh. 6.4 - Prob. 22ECh. 6.4 - Suppose A = QR, where Q is m n and R is n n....Ch. 6.4 - Suppose A = QR, where R is an invertible matrix....Ch. 6.4 - Given A = QR as in Theorem 12, describe how to...Ch. 6.4 - Let u1, , up be an orthogonal basis for a subspace...Ch. 6.4 - Suppose A = QR is a QR factorization of an m n...Ch. 6.4 - [M] Use the Gram-Schmidt process as in Example 2...Ch. 6.5 - Let A = [133151172] and b = [535]. Find a...Ch. 6.5 - What can you say about the least-squares solution...Ch. 6.5 - In Exercises 1-4, find a least-squares solution of...Ch. 6.5 - In Exercises 1-4, find a least-squares solution of...Ch. 6.5 - In Exercises 1-4, find a least-squares solution of...Ch. 6.5 - In Exercises 1-4, find a least-squares solution of...Ch. 6.5 - In Exercises 5 and 6, describe all least-squares...Ch. 6.5 - In Exercises 5 and 6, describe all least-squares...Ch. 6.5 - Compute the least-squares error associated with...Ch. 6.5 - Compute the least-squares error associated with...Ch. 6.5 - In Exercises 9-12, find (a) the orthogonal...Ch. 6.5 - In Exercises 9-12, find (a) the orthogonal...Ch. 6.5 - In Exercises 9-12, find (a) the orthogonal...Ch. 6.5 - In Exercises 9-12, find (a) the orthogonal...Ch. 6.5 - Let A = [342134], b = [1195], u = [51], and v =...Ch. 6.5 - Let A = [213432], b = [544], u = [45], and v =...Ch. 6.5 - In Exercises 15 and 16, use the factorization A =...Ch. 6.5 - In Exercises 15 and 16, use the factorization A =...Ch. 6.5 - In Exercises 17—26, A is an mn matrix and b is...Ch. 6.5 - Prob. 18ECh. 6.5 - Prob. 19ECh. 6.5 - Prob. 20ECh. 6.5 - Prob. 21ECh. 6.5 - Prob. 22ECh. 6.5 - Prob. 23ECh. 6.5 - Prob. 24ECh. 6.5 - Prob. 25ECh. 6.5 - Prob. 26ECh. 6.5 - Let A be an m n matrix. Use the steps below to...Ch. 6.5 - Let A be an m n matrix such that ATA is...Ch. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.5 - Suppose A is m n with linearly independent...Ch. 6.5 - Find a formula for the least-squares solution of...Ch. 6.5 - Describe all least-squares solutions of the system...Ch. 6.6 - When the monthly sales of a product are subject to...Ch. 6.6 - In Exercises 1-4, find the equation y = 0 + 1x of...Ch. 6.6 - In Exercises 1-4, find the equation y = 0 + 1x of...Ch. 6.6 - In Exercises 1-4, find the equation y = 0 + 1x of...Ch. 6.6 - In Exercises 1-4, find the equation y = 0 + 1x of...Ch. 6.6 - Prob. 5ECh. 6.6 - Prob. 6ECh. 6.6 - If a machine learns the least-squares line that...Ch. 6.6 - Prob. 8ECh. 6.6 - Prob. 9ECh. 6.6 - Prob. 10ECh. 6.6 - Let X be the design matrix used to find the...Ch. 6.6 - Let X be the design matrix in Example 2...Ch. 6.6 - Prob. 13ECh. 6.6 - Prob. 14ECh. 6.6 - A certain experiment produces the data (1, 7.9),...Ch. 6.6 - Let x=1n(x1++xn) and y=1n(y1++yn). Show that the...Ch. 6.6 - Derive the normal equations (7) from the matrix...Ch. 6.6 - Use a matrix inverse to solve the system of...Ch. 6.6 - a. Rewrite the data in Example 1 with new...Ch. 6.6 - Suppose the x-coordinates of the data (x1, y1), ,...Ch. 6.6 - Exercises 19 and 20 involve a design matrix X with...Ch. 6.6 - Show that X2=TXTy. [Hint: Rewrite the left side...Ch. 6.7 - Use the inner product axioms to verify the...Ch. 6.7 - Use the inner product axioms to verify the...Ch. 6.7 - Let 2 have the inner product of Example 1, and let...Ch. 6.7 - Let 2 have the inner product of Example 1. Show...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Exercises 3-8 refer to 2 with the inner product...Ch. 6.7 - Let 3 have the inner product given by evaluation...Ch. 6.7 - Let 3 have the inner product as in Exercise 9,...Ch. 6.7 - Let p0, p1, and p2 be the orthogonal polynomials...Ch. 6.7 - Find a polynomial p3 such that {p0, p1, p2, p3}...Ch. 6.7 - Let A be any invertible n n matrix. Show that for...Ch. 6.7 - Let T be a one-to-one linear transformation from a...Ch. 6.7 - Use the inner product axioms and other results of...Ch. 6.7 - Use the inner product axioms and other results of...Ch. 6.7 - Use the inner product axioms and other results of...Ch. 6.7 - Use the inner product axioms and other results of...Ch. 6.7 - Prob. 19ECh. 6.7 - Prob. 20ECh. 6.7 - Prob. 21ECh. 6.7 - Prob. 22ECh. 6.7 - Prob. 23ECh. 6.7 - Prob. 24ECh. 6.7 - Given a 0 and b 0, let u=[ab] and v=[ba]. Use...Ch. 6.7 - Let u=[ab] and v=[11]. Use the Cauchy-Schwarz...Ch. 6.7 - Exercises 21-24 refer to V = C[0, 1], with the...Ch. 6.7 - Exercises 21-24 refer to V = C[0, 1], with the...Ch. 6.7 - Prob. 29ECh. 6.7 - Prob. 30ECh. 6.7 - Let V be the space C[1, 1] with the inner product...Ch. 6.7 - Let V be the space C[2, 2] with the inner product...Ch. 6.8 - Let q1(t) = 1, q2(t) = t, and q3(t) = 3t2 4....Ch. 6.8 - Find the first-order and third-order Fourier...Ch. 6.8 - Find the least-squares line y = 0 + 1x that best...Ch. 6.8 - Suppose 5 out of 25 data points in a weighted...Ch. 6.8 - Fit a cubic trend function to the data in Example...Ch. 6.8 - To make a trend analysis of six evenly spaced data...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - Prob. 7ECh. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - In Exercises 5-14, the space is C[0, 2] with the...Ch. 6.8 - [M] Let f4 and f5 be the fourth-order and...Ch. 6 - Prob. 1SECh. 6 - Prob. 2SECh. 6 - Prob. 3SECh. 6 - Prob. 4SECh. 6 - Prob. 5SECh. 6 - Prob. 6SECh. 6 - Prob. 7SECh. 6 - Prob. 8SECh. 6 - Prob. 9SECh. 6 - Prob. 10SECh. 6 - Prob. 11SECh. 6 - Prob. 12SECh. 6 - Prob. 13SECh. 6 - Prob. 14SECh. 6 - The statements in Exercises 1—19 refer to...Ch. 6 - Prob. 16SECh. 6 - Prob. 20SECh. 6 - Let {v1, , vp} be an orthonormal set in n. Verify...Ch. 6 - Let U be an n n orthogonal matrix. Show that if...Ch. 6 - Show that if an n n matrix U satisfies (Ux) (Uy)...Ch. 6 - Show that if U is an orthogonal matrix, then any...Ch. 6 - A Householder matrix, or an elementary reflector,...Ch. 6 - Let T: n n be a linear transformation that...Ch. 6 - Let u and v be linearly independent vectors in n...Ch. 6 - Suppose the columns of A are linearly independent....Ch. 6 - If a, b, and c are distinct numbers, then the...Ch. 6 - Consider the problem of finding an eigenvalue of...Ch. 6 - Use the steps below to prove the following...Ch. 6 - Explain why an equation Ax = b has a solution if...Ch. 6 - Exercises 15 and 16 concern the (real) Schur...Ch. 6 - Let A be an n n matrix with n real eigenvalues,...
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