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 7, Problem 9CTA
To determine
True or false: the Gram-Schmidt and Householder QR factorizations, and the singular value decomposition, will all compute highly accurate solutions.
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5. Find the QR factorization of the matrix and use it to solve the least-squares problem:
1
X1
X2
=
10
Find the norm of the residual at the least-squares solution.
Suppose that Ax=b has no solution and the columns of A are linearly independent. Let t(A) denote
the transpose of matrix A.
Select all true statements.
O If the A matrix and b vector represent data values, then the least-squares solution gives coefficients for a
function that make this function the line that best fits the data values.
O The system has a least-squares solution.
The solution to t(A)Ax=t(A)b is the least-squares solution.
O In the system Ax-b=e, it is possible for e to be the zero vector.
O The system has a least squares solution but it may not be unique.
O The inverse of t(A)A exists.
O The least-squares solution (if it exists) yields a vector in the column space of A that is the "closest" to the
vector b.
Find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of Ax = b.
4
1 -9
1
A =
10
1
1 -1 -9
a. The orthogonal projection of b onto Col A is 6 =D (Simplify your answer.)
Chapter 7 Solutions
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Ch. 7.1 - Find the three-digit decimal floating-point...Ch. 7.1 - Prob. 2ECh. 7.1 - Represent each of the following numbers as...Ch. 7.1 - Prob. 4ECh. 7.1 - Prob. 5ECh. 7.1 - Prob. 6ECh. 7.1 - Prob. 7ECh. 7.1 - Prob. 8ECh. 7.1 - Prob. 9ECh. 7.1 - Prob. 10E
Ch. 7.2 - Let A=(111241 31 2) Factor A into a product LU,...Ch. 7.2 - Prob. 2ECh. 7.2 - Let A and B be nn matrices and let xn. How many...Ch. 7.2 - Let Amn,Bnr, and x, yn. Suppose that the product...Ch. 7.2 - Let Eki be the elementary matrix formed by...Ch. 7.2 - Prob. 6ECh. 7.2 - If A is a symmetric nn matrix with triangular...Ch. 7.2 - Prob. 8ECh. 7.2 - Let A=LU, where L is lower triangular with 1's on...Ch. 7.2 - Suppose that A1 and the LU factorization of A have...Ch. 7.2 - Prob. 11ECh. 7.3 - Let A=(03112 2254) and b=(17 1) Reorder the rows...Ch. 7.3 - Let A be the matrix in Exercise 1. Use the...Ch. 7.3 - Prob. 3ECh. 7.3 - Prob. 4ECh. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Prob. 7ECh. 7.3 - Prob. 8ECh. 7.3 - Solve the system in Exercise 7 using four-digit...Ch. 7.3 - Use four-digit decimal floating-point arithmetic,...Ch. 7.4 - Determine F,, and 1 for each of the following...Ch. 7.4 - Let A=(200 2) and x=( x 1 x 2 ) and set...Ch. 7.4 - Let A=(1000) Use the method of Exercise 2 to...Ch. 7.4 - Let D=(30000 50000 200004) Compute the singular...Ch. 7.4 - Prob. 5ECh. 7.4 - If D is an nn diagonal matrix, how do the values...Ch. 7.4 - Prob. 7ECh. 7.4 - Let M denote a matrix norm on nn,V denote a vector...Ch. 7.4 - A vector x in n can also be viewed as an n1 matrix...Ch. 7.4 - A vector y in n can also be viewed as an n1 matrix...Ch. 7.4 - Let A=wyT where wm and yn. Show that Ax2x2y2w2 for...Ch. 7.4 - Prob. 12ECh. 7.4 - Theorem 7.4.2 status that A=max1im(j=1n| a ij|)...Ch. 7.4 - Prob. 14ECh. 7.4 - Prob. 15ECh. 7.4 - Prob. 16ECh. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - Prob. 19ECh. 7.4 - Prob. 20ECh. 7.4 - Let A be an mn matrix. Show that A(1,2)A2Ch. 7.4 - Let Amn and Bnr . Show that Ax2A(1,2)x1 for all x...Ch. 7.4 - Let A be an nn matrix and let m be a matrix norm...Ch. 7.4 - Prob. 24ECh. 7.4 - Prob. 25ECh. 7.4 - Prob. 26ECh. 7.4 - Let A be an nn matrix and xn. Prove: Axn1/2A2x...Ch. 7.4 - Prob. 28ECh. 7.4 - Prob. 29ECh. 7.4 - Solve the given two systems and compare the...Ch. 7.4 - Prob. 31ECh. 7.4 - Prob. 32ECh. 7.4 - Let An=(111 1 1 n ) for each positive integer n....Ch. 7.4 - Prob. 34ECh. 7.4 - Given A=(3211) and b=(52) If two-digit decimal...Ch. 7.4 - Prob. 36ECh. 7.4 - Prob. 37ECh. 7.4 - Prob. 38ECh. 7.4 - Let A and B be nonsingular nn matrices. Show that...Ch. 7.4 - Prob. 40ECh. 7.4 - Prob. 41ECh. 7.4 - Let A be an nn matrix and let Q and V be nn...Ch. 7.4 - Prob. 43ECh. 7.4 - Prob. 44ECh. 7.4 - Let A be an mn matrix with singular value...Ch. 7.4 - Let A be a nonsingular nn matrix and let Q be an...Ch. 7.4 - Let A be a symmetric nonsingular nn matrix with...Ch. 7.5 - For each of the following vectors x, find a...Ch. 7.5 - Given x3, define rij=(xi2+xj2)1/2i,j=1,2,3 For...Ch. 7.5 - For each of the given vectors x, find a...Ch. 7.5 - For each of the following, find a Householder...Ch. 7.5 - Prob. 5ECh. 7.5 - Let A=( 1 3 2 1 2 288 2 71) and b=( 11 2 01) Use...Ch. 7.5 - Prob. 7ECh. 7.5 - Prob. 8ECh. 7.5 - Let Hk=I2uuT be a Householder transformation with...Ch. 7.5 - Let QT=GnkG2G1, where each Gi is a Givens...Ch. 7.5 - Prob. 11ECh. 7.5 - Prob. 12ECh. 7.5 - Prob. 13ECh. 7.5 - Let R be an nn plane rotation. What is the value...Ch. 7.5 - Prob. 15ECh. 7.5 - Prob. 16ECh. 7.5 - Prob. 17ECh. 7.6 - Let A=(1111) Apply one iteration of the power...Ch. 7.6 - Let A=(210131012) and u0=(111) Apply the power...Ch. 7.6 - Let A=(12 1 1) and u0=(11) Compute u1,u2,u3, and...Ch. 7.6 - Let A=A1=(1113) Compute A2 and A3, using the QR...Ch. 7.6 - Let A=(522 21 2 3 42) Verify that 1=4 is an...Ch. 7.6 - Let A be an nn matrix with distinct real...Ch. 7.6 - Prob. 7ECh. 7.6 - Prob. 8ECh. 7.6 - Prob. 9ECh. 7.6 - Prob. 10ECh. 7.6 - Prob. 11ECh. 7.6 - Prob. 12ECh. 7.6 - Let R be an nn upper triangular matrix whose...Ch. 7.7 - Prob. 1ECh. 7.7 - Prob. 2ECh. 7.7 - Let A=(10131310),b=( 4222) Use Householder...Ch. 7.7 - Prob. 4ECh. 7.7 - Let A=(1100) where is a small scalar. Determine...Ch. 7.7 - Show that the pseudoinverse A+ satisfies the four...Ch. 7.7 - Prob. 7ECh. 7.7 - Prob. 8ECh. 7.7 - Show that if A is a mn matrix of rank n, then...Ch. 7.7 - Prob. 10ECh. 7.7 - Prob. 11ECh. 7.7 - Let A=(111100) Determine A+ and verify that A and...Ch. 7.7 - Let A=(12 1 2) and b=(6 4) Compute the singular...Ch. 7.7 - Prob. 14ECh. 7.7 - Prob. 15ECh. 7.7 - Prob. 16ECh. 7 - Set A=round(10*rand(6))s=ones(6,1)b=A*s The...Ch. 7 - Prob. 2ECh. 7 - Prob. 3ECh. 7 - Prob. 4ECh. 7 - Prob. 5ECh. 7 - Prob. 6ECh. 7 - Prob. 7ECh. 7 - Prob. 8ECh. 7 - Construct a matrix A as follows: A=diag(11:1:1,1);...Ch. 7 - Prob. 10ECh. 7 - Set x1=(1:5);x2=[1,3,4,5,9];x=[x1;x2] Construct a...Ch. 7 - To plot y=sin(x), we must define vectors of x and...Ch. 7 - Let A=(452452036036) Enter the matrix A in MATLAB...Ch. 7 - Set A=round(10*rand(10,5)) and s=svd(A) Use MATLAB...Ch. 7 - Prob. 15ECh. 7 - Prob. 16ECh. 7 - Prob. 17ECh. 7 - Prob. 18ECh. 7 - Prob. 19ECh. 7 - Prob. 1CTACh. 7 - Prob. 2CTACh. 7 - If A is a nonsingular matrix and a numerically...Ch. 7 - If A is a symmetric matrix and a numerically...Ch. 7 - Prob. 5CTACh. 7 - Prob. 6CTACh. 7 - If A is a symmetric matrix, then A1=A.Ch. 7 - Prob. 8CTACh. 7 - Prob. 9CTACh. 7 - Prob. 10CTACh. 7 - Prob. 1CTBCh. 7 - Let A=(236448134)b=(304)c=(182) Use Gaussian...Ch. 7 - Prob. 3CTBCh. 7 - Prob. 4CTBCh. 7 - Let A be a 1010 matrix with cond(A)=5106 . Suppose...Ch. 7 - Prob. 6CTBCh. 7 - Prob. 7CTBCh. 7 - Prob. 8CTBCh. 7 - Let A=(524524360360) and b=(51 19) The singular...Ch. 7 - Prob. 10CTB
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