Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
9th Edition
ISBN: 9780321962218
Author: Steven J. Leon
Publisher: PEARSON
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Chapter 5.4, Problem 16E
Let
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The value y (in 1982–1984 dollars) of each dollar paid by consumers in each of the years from 1994 through 2008 in a country is represented by the ordered pairs.
(1994, 0.677)
(1995, 0.655)
(1996, 0.639)
(1997, 0.618)
(1998, 0.609)
(1999, 0.605)
(2000, 0.580)
(2001, 0.565)
(2002, 0.557)
(2003, 0.542)
(2004, 0.531)
(2005, 0.516)
(2006, 0.499)
(2007, 0.478)
(2008, 0.461)
(a) Use a spreadsheet software program to generate a scatter plot of the data. Let t = 4 represent 1994. Do the data appear linear? Yes or No (b) Use the regression feature of the spreadsheet software program to find a linear model for the data. (Let t represent time. Round your numerical values to four decimal places.)
y =
(c) Use the model to predict the value (in 1982–1984 dollars) of 1 dollar paid by consumers in 2010 and in 2013. (Round your answers to two decimal places.)
2010
$
2013
$
Discuss the…
A transportation engineer observed that in five zones of the city, the number of gas stations (Y) in relation to the population (X) in 1000's was: (as shown in the table). Determine the model representing the change of each variable and explain the relationship if it is linear
X
1
5
3
2
4
Y
2
7
3
5
8
Determine the quadratic equation that best fits the given set of data:
(1, 2), (2, 3), (4, 3), (6,4), (7, 8)
310
273
y=.
-
318
91
318
91
310
-+-
91
318
21
310
273
x +
·x +
x +
310 5
273
+x+
21
5
-1-²
21
5
-1.²
273
5
=x²
21
318
91
Chapter 5 Solutions
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Ch. 5.1 - Find the angle between the vectors v and w in each...Ch. 5.1 - For each pair of vectors in Exercise 1, find...Ch. 5.1 - For each of the following pairs of vectors x and...Ch. 5.1 - Let x and y be linearly independent vectors in 2....Ch. 5.1 - Find the point on the line y=2x that is closer to...Ch. 5.1 - Find the point on the line y=2x+1 that is closet...Ch. 5.1 - Find the distance from the point (1, 2) to the...Ch. 5.1 - In each of the following, find the equation of the...Ch. 5.1 - Find the equation of the plane that passes through...Ch. 5.1 - Find the distance from the point (1,1,1) to be...
Ch. 5.1 - Findthedistancefromthepoint (2,1,2) totheplane...Ch. 5.1 - If x=(x1,x2)T,y=(y1,y2)T, and z=(z1,z2)T...Ch. 5.1 - Prob. 13ECh. 5.1 - Let x1,x2, and x3 be vectors in 3. If x1x2 and...Ch. 5.1 - Let A be a 22 matrix with linearly independent...Ch. 5.1 - If x and y are linearly independent vectors in 3,...Ch. 5.1 - Let x=(44 44) and y=(4221) Determine the angle...Ch. 5.1 - Let x and y be vectors in n and define p=xTyyTyy...Ch. 5.1 - Use the database matrix U from Application 1 and...Ch. 5.1 - Fivestudentsinanelementaryschooltakeaptitude tests...Ch. 5.1 - Let t be a fixed real number and let...Ch. 5.2 - For each of the following matrices, determine a...Ch. 5.2 - Let S be the subspace of 3 spanned by x=(1,1)T....Ch. 5.2 - a.Let S be the subspace of 3 spanned by the...Ch. 5.2 - Let S be the subspace of 4 spanned by...Ch. 5.2 - Let A be a 32 matrix with rank 2. Give geometric...Ch. 5.2 - Is it possible for a matrix to have the vector...Ch. 5.2 - Let aj be a nonzero column vector of an mn matrix...Ch. 5.2 - Let S be the subspace of n spanned by the vectors...Ch. 5.2 - If A is an mn matrix of rank r, what are the...Ch. 5.2 - Prob. 10ECh. 5.2 - Prove: If A is an mn matrix and xn, then either...Ch. 5.2 - Let A be an mn matrix. Explain why the following...Ch. 5.2 - Let A bean mn matrix.Showthat If xN(ATA), then Ax...Ch. 5.2 - Let A be an mn matrix, B an nr matrix, and C=AB....Ch. 5.2 - Let U and V be subspaces of a vector space W. Show...Ch. 5.2 - Let A be an mn matrix of rank r and let...Ch. 5.2 - Let x and y be linearly independent vectors in n...Ch. 5.3 - Find the least squares solution of each of the...Ch. 5.3 - For each of your solutions x in Exercise 1:...Ch. 5.3 - For each of the following systems Ax=b, find...Ch. 5.3 - ForeachofthesystemsinExercise3,determinethe...Ch. 5.3 - Find the best least squares fit by a linear...Ch. 5.3 - Find the best least squares fit to the data in...Ch. 5.3 - Given a collection of points...Ch. 5.3 - The point (x,y) is the center of mass for the...Ch. 5.3 - LetAbean mnmatrixofranknandletP=A(ATA)1AT. (a)...Ch. 5.3 - LetAbean 85 matrixofrank3,andletbbea nonzero...Ch. 5.3 - Let P=A(ATA)1AT, where A is an mn matrixof rank n....Ch. 5.3 - Show that if (AIO A T )( x r)=(b0) then x is a...Ch. 5.3 - Let and let be a solution of the leastsquares...Ch. 5.3 - Find the equation of the circle that gives the...Ch. 5.3 - Prob. 15ECh. 5.4 - Let x=(1,1,1,1)T and y=(1,1,5,3)T. Showthat xy....Ch. 5.4 - Let x=(1,1,1,1)T and y=(8,2,2,0)T....Ch. 5.4 - Use equation (1) with weight vector w=(14,12,14)T...Ch. 5.4 - Given A=(122102311) and B=( 411 3321 2 2)...Ch. 5.4 - Show that equation (2) defines an inner product on...Ch. 5.4 - Showthattheinnerproductdefinedbyequation(3)...Ch. 5.4 - In C[0,1], with inner product defined by (3),...Ch. 5.4 - In C[0,1], with inner product defined by (3),...Ch. 5.4 - In C[,] with inner product defined by (6), show...Ch. 5.4 - Show that the functions x and x2 are orthogonal in...Ch. 5.4 - In P5 with inner product as in Exercise 10 and...Ch. 5.4 - If V is an inner product space, show that v=v,v...Ch. 5.4 - Show that x1=i=1n|xi| defines a norm on n.Ch. 5.4 - Show that x=max1in|xi| defines a norm on n.Ch. 5.4 - Compute x1,x2, and x for each of the following...Ch. 5.4 - Let x=(5,2,4)T and y=(3,3,2)T. Compute xy1,xy2,...Ch. 5.4 - Prob. 17ECh. 5.4 - Prob. 18ECh. 5.4 - In n with inner product x,y=xTy Derive a formula...Ch. 5.4 - Prob. 20ECh. 5.4 - Let xn. Show that xx2.Ch. 5.4 - Prob. 22ECh. 5.4 - Prob. 23ECh. 5.4 - Prob. 24ECh. 5.4 - Prob. 25ECh. 5.4 - Prove that, for any u and v in an inner...Ch. 5.4 - The result of Exercise 26 is not valid for norms...Ch. 5.4 - Determine whether the following define norms on...Ch. 5.4 - Let xn and show that x1nx x2nx Give examples of...Ch. 5.4 - Sketch the set of points (x1,x2)=xT in 2 such that...Ch. 5.4 - LetK bean nn matrixoftheform K=(1 c c c0s sc sc00...Ch. 5.4 - Thetraceofan nn matrixC, denoted tr(C), isthe sum...Ch. 5.4 - Consider the vector space n with inner product...Ch. 5.5 - Which of the following sets of vectors form an...Ch. 5.5 - Let u1=( 1 3 2 1 3 2 4 3 2 ),u2=( 2 3 2 3 1 3...Ch. 5.5 - Let S be the subspace of 3 spanned by the vectors...Ch. 5.5 - Let be a fixed real number and let x1=( cos sin)...Ch. 5.5 - Let u1 and u2 form an orthonormal basis for 2 and...Ch. 5.5 - Let {u1,u2,u3} be an orthonormal basis for an...Ch. 5.5 - Let {u1,u2,u3} beanorthonormalbasisforaninner...Ch. 5.5 - The functions cosx and sinx form an orthonormal...Ch. 5.5 - The set S={12,cosx,cos2x,cos3x,cos4x}...Ch. 5.5 - Prob. 10ECh. 5.5 - Prob. 11ECh. 5.5 - If Q is an nn orthogonal matrix and x and y are...Ch. 5.5 - Prob. 13ECh. 5.5 - Prob. 14ECh. 5.5 - Let Q be an orthogonal matrix and let d=det(Q)....Ch. 5.5 - Show that the product of two orthogonal matrices...Ch. 5.5 - Prob. 17ECh. 5.5 - Prob. 18ECh. 5.5 - Prob. 19ECh. 5.5 - Prob. 20ECh. 5.5 - Let A=( 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ) Show...Ch. 5.5 - Prob. 22ECh. 5.5 - Prob. 23ECh. 5.5 - Let A be an mn matrix, let P be the projection...Ch. 5.5 - Let P be the projection matrix corresponding to a...Ch. 5.5 - Prob. 26ECh. 5.5 - Let v be a vector in an inner product space V...Ch. 5.5 - Let v be a vector in an inner product space V and...Ch. 5.5 - Given the vector space C[1,1] with inner product...Ch. 5.5 - Consider the inner product space C[0,1] with inner...Ch. 5.5 - Prob. 31ECh. 5.5 - Find the best least squares approximation to...Ch. 5.5 - Let {x1,x2,...,xk,xk+1,...,xn} be an orthonormal...Ch. 5.5 - Prob. 34ECh. 5.5 - Prob. 35ECh. 5.5 - A(real or complex)scalar u is said to bean nth...Ch. 5.5 - Prob. 37ECh. 5.5 - Prob. 38ECh. 5.6 - For each of the following, use the GramSchmidt...Ch. 5.6 - Factor each of the matrices in Exercise 1 into a...Ch. 5.6 - Giventhebasis {(1,2,2)T,(1,2,1)T} for 3, use the...Ch. 5.6 - Consider the vector space C[1,1] with innerproduct...Ch. 5.6 - Let A=(211121) and b=( 126 18) Use the GramSchmidt...Ch. 5.6 - Repeat Exercises 5 using A=(3 14202) and b=(0 20...Ch. 5.6 - Given x1=12(1,1,1,1)T and x2=16(1,1,3,5)T, verify...Ch. 5.6 - Use the GramSchmidt process to find an orthonormal...Ch. 5.6 - Repeat Exercise 8 using the modified GramSchmidt...Ch. 5.6 - Let A be an m2 matrix. Show that if both the...Ch. 5.6 - LetAbean m3 matrix.LetQRbetheQRfactorization...Ch. 5.6 - What will happen if the GramSchmidt process is...Ch. 5.6 - Let Abeanmn matrix of rank n and let bm. Show that...Ch. 5.6 - Let U be an m-dimensional subspace of n and let V...Ch. 5.6 - (Dimension Theorem) Let U and V be subspaces of n....Ch. 5.7 - Use the recursion formulas to calculate (a) T4,T5...Ch. 5.7 - Prob. 2ECh. 5.7 - Prob. 3ECh. 5.7 - Prob. 4ECh. 5.7 - Prob. 5ECh. 5.7 - Prob. 6ECh. 5.7 - Prob. 7ECh. 5.7 - Prob. 8ECh. 5.7 - Prob. 9ECh. 5.7 - Prove each of the following....Ch. 5.7 - Givenafunction f(x) thatpassesthroughthepoints...Ch. 5.7 - Prob. 12ECh. 5.7 - Prob. 13ECh. 5.7 - Prob. 14ECh. 5.7 - Let x1,x2,...,xn be distinct point in the interval...Ch. 5.7 - Prob. 16ECh. 5.7 - Prob. 17ECh. 5 - Set x=[0:4,4,1,1] and y=ones(9,1) Use the MATLAB...Ch. 5 - Prob. 2ECh. 5 - Prob. 3ECh. 5 - (Least Squares Circles) The parametric equations...Ch. 5 - Prob. 5ECh. 5 - Prob. 1CTACh. 5 - If x and y are unit vectors in n and |xTy|=1, then...Ch. 5 - If U, V, and W are subspaces of 3 and if UV and...Ch. 5 - It is possible to find a nonzero vector y in the...Ch. 5 - Prob. 5CTACh. 5 - Prob. 6CTACh. 5 - If N(A)={0}, then the system Ax=b will have a...Ch. 5 - Prob. 8CTACh. 5 - Prob. 9CTACh. 5 - Prob. 10CTACh. 5 - Prob. 1CTBCh. 5 - Prob. 2CTBCh. 5 - Prob. 3CTBCh. 5 - Let A be a 75 matrix with rank equal to 4 and let...Ch. 5 - Letxandybevectorsin n andletQbean nn orthogonal...Ch. 5 - Let S be the two-dimensional subspace of 3 spanned...Ch. 5 - Prob. 7CTBCh. 5 - Prob. 8CTBCh. 5 - Prob. 9CTBCh. 5 - Prob. 10CTBCh. 5 - The functions cosx and sinx are both unit vectors...Ch. 5 - Prob. 12CTB
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- Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4, u5 and u6. Then verify your solution. v=(10,30,13,14,7,27) u1=(1,2,3,4,1,2) u2=(1,2,1,1,2,1) u3=(0,2,1,2,1,1) u4=(1,0,3,4,1,2) u5=(1,2,1,1,2,3) u6=(3,2,1,2,3,0)arrow_forwardThe prices (in $) of Rawlston, Inc. stock ( y) over a period of 12 days, the number of shares (in 100s) of company's stocks sold ( x 1), and the volume of exchange (in millions) on the New York Stock Exchange ( x 2) are shown below. Day (y) (x1) (x2) 1 87.50 950 11.00 2 86.00 945 11.25 3 84.00 940 11.75 4 83.00 930 11.75 5 84.50 935 12.00 6 84.00 935 13.00 7 82.00 932 13.25 8 80.00 938 14.50 9 78.50 925 15.00 10 79.00 900 16.50 11 77.00 875 17.00 12 77.50 870 17.50 Excel was used to determine the least squares regression equation. Part of the computer output is shown below. ANOVA df SS MS F Significance F Regression 2 118.8474 59.4237 40.9216 0.0000 Residual 9 13.0692 1.4521 Total 11 131.9167 Coefficients Standard Error t Stat P-value Intercept…arrow_forwardThe value y (in 1982–1984 dollars) of each dollar paid by consumers in each of the years from 1994 through 2008 in a country is represented by the ordered pairs. (1994, 0.675) (1995, 0.654) (1996, 0.643) (1997, 0.618) (1998, 0.611) (1999, 0.599) (2000, 0.585) (2001, 0.562) (2002, 0.559) (2003, 0.539) (2004, 0.529) (2005, 0.513) (2006, 0.493) (2007, 0.482) (2008, 0.462) (b) Use the regression feature of the spreadsheet software program to find a linear model for the data. (Let trepresent time. Round your numerical values to four decimal places.) y = (c) Use the model to predict the value (in 1982–1984 dollars) of 1 dollar paid by consumers in 2010 and in 2011. (Round your answers to two decimal places.)arrow_forward
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