Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
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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Textbook Question
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Chapter 3, Problem 4E

(Rank−1 Updates of Linear Systems)
(a) Set
A = round ( 10 rand ( 8 ) )
b = round ( 10 rand ( 8 , 1 ) )
M = inv ( A )
Use the matrix M to solve the system A y = b for y.
(b) Consider now a new system C x = b , where C is constructed as follows:
u = round ( 10 rand ( 8 , 1 ) )
v = round ( 10 rand ( 8 , 1 ) )
E = u v
C = A + E
The matrices C and A differ by the rank−1 matrix E. Use MATLAB to verify that the rank of E is 1. Use MATLABS’s “\” operation to solve the system C x = b and then compute the residual vector r = b A x .
(c) Let us now solve C x = b by a new method that takes advantage of the fact that A and C differ by a rank−1 matrix. This new procedure is called a rank−1 update method. Set
z = M u , c = v y
d = v z, e = c / ( 1 + d )
and then compute the solution x by
x = y e z
Compute the residual vector b C x and compare it with the residual vector in part (b). This new method may seem more complicated, but it actually is much more computationally efficient.
(d) To see why the rank−1 update method works, use MATLAB to compute and compare
Cy and b + c u
Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Also, compute
Cz and ( 1 + d ) u
Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Use these identities to prove that C x = b . Assuming that A is nonsingular, will the rank−1 update method always work? Under what conditions could it fail? Explain.

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Chapter 3 Solutions

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Ch. 3.1 - Let V be the set of all ordered pairs of real...Ch. 3.1 - Let R+ denote the set of positive real numbers....Ch. 3.1 - Let R denote the set of real numbers. Define...Ch. 3.1 - Let Z denote the set of ail integers with addition...Ch. 3.1 - LetS denote the set of all infinite sequences of...Ch. 3.1 - We can define a onetoone correspondence between...Ch. 3.2 - Determine whether the following sets form...Ch. 3.2 - Determine whether the following sets form sub...Ch. 3.2 - Determine whether the following are subspaces of...Ch. 3.2 - Determine the null space of each of the following...Ch. 3.2 - Determine whether the following are subspaces of...Ch. 3.2 - Determine whether the following are subspaces of...Ch. 3.2 - Show that Cn[a,b] is a subspace of C[a,b] .Ch. 3.2 - Let A be a fixed vector in nnandletSbethesetof all...Ch. 3.2 - In each of the following determine the subspace of...Ch. 3.2 - LetA be a particular vector in 22 ....Ch. 3.2 - Determine whether the following are spanning...Ch. 3.2 - Which of the sets that follow are spanning sets...Ch. 3.2 - Given x1=(123),x2=(342) x=(266),y=(925) Is...Ch. 3.2 - Let A be a 43 matrixand let b4 . Howmanypossible...Ch. 3.2 - Let A be a 43 matrixandlet c=2a1+a2+a3 (a) If...Ch. 3.2 - Let x1 be a particular solution to a system Ax=b...Ch. 3.2 - Let {x1,x2,...xk} be a spanning set for a vector...Ch. 3.2 - In 22 , let E11=(1000),E12=(0100)...Ch. 3.2 - Prob. 19ECh. 3.2 - Let S be the vector space of infinite...Ch. 3.2 - Prove that if S is a subspace of 1 , then either...Ch. 3.2 - Let Abe an nn matrix. Prove that the...Ch. 3.2 - Let U and V be subspaces of a vector space W.Prove...Ch. 3.2 - Let S be the subspace of 2 spanned by e1 and letT...Ch. 3.2 - Let U and V be subspaces of a vector space W....Ch. 3.2 - Let S, T, and U be subspaces of a vector space V....Ch. 3.3 - Determine whether the following vectors are...Ch. 3.3 - Determine whether the following vectors are...Ch. 3.3 - For each of the sets of vectors in Exercise 2,...Ch. 3.3 - Determine whether the following vectors are...Ch. 3.3 - Let x1,x2,...,xk be linearly independent vectors...Ch. 3.3 - Let x1,x2 , and x3 be linearly independent vectors...Ch. 3.3 - Let x1,x2 , and x3 be linearly independent vectors...Ch. 3.3 - Determine whether the following vectors are...Ch. 3.3 - Prob. 9ECh. 3.3 - Determine whether the vectors cosx,1 , and...Ch. 3.3 - Consider the vectors cos(x+) and sinx in C[,] ....Ch. 3.3 - Given the functions 2x and |x| , show that (a)...Ch. 3.3 - Prove that any finite set of vectors that contains...Ch. 3.3 - Let v1 and v2 be two vectors in a vector space...Ch. 3.3 - Prove that any nonempty subset of a linearly...Ch. 3.3 - Let Abe an mn matrix. Show that if A has linearly...Ch. 3.3 - Let x1,...,xk be linearly independent vectors in n...Ch. 3.3 - Prob. 18ECh. 3.3 - Prob. 19ECh. 3.3 - Let v1,v2,...,vn be linearly independent vectorsin...Ch. 3.4 - In Exercise 1 of Section 3.3, indicate whether the...Ch. 3.4 - In Exercise 2 of Section 3.3, indicate whether the...Ch. 3.4 - Consider the vectors x1=(21),x2=(43),x3=(73) (a)...Ch. 3.4 - Given the vectors x1=(324),x2=(324),x3=(648) what...Ch. 3.4 - Let x1=(213),x2=(314),x3=(264) (a) Show that x1,x2...Ch. 3.4 - In Exercise 2 of Section 3.2, some of the sets...Ch. 3.4 - Find a basis for the subspace S of 4 consisting of...Ch. 3.4 - Given x1=(1,1,1)T and x2=(3,1,4)T : (a) Do x1 and...Ch. 3.4 - Let a1 and a2 be linearly independent vectors in 3...Ch. 3.4 - The vectors x1=(122) , x2=(254) , x3=(132) ,...Ch. 3.4 - Let S be the subspace of P3 consisting of all...Ch. 3.4 - In Exercise 3 of Section 3.2, some of the sets...Ch. 3.4 - In C[,] , find the dimension of the...Ch. 3.4 - In each of the following, find the dimension of...Ch. 3.4 - LetS be the subspace of P3 consisting of all...Ch. 3.4 - In 4 let U be the subspace of all vectors of the...Ch. 3.4 - Is it possible to find a pair of twodimensional...Ch. 3.4 - Show that if U and V are subspaces of n and UV=0 ,...Ch. 3.5 - For each of the following, find the transition...Ch. 3.5 - For each of the ordered bases u1,u2 in Exercise 1,...Ch. 3.5 - Let v1(3,2)T and v2(4,3)T . For each orderedbasis...Ch. 3.5 - Let E=[(5,3)T,(3,2)T] and let x=(1,1)T , y=(1,1)T...Ch. 3.5 - Let u1=(1,1,1)T,u2=(1,2,2)T , and u3=(2,3,4)T (a)...Ch. 3.5 - Let v1=(4,6,7)T,v2=(0,1,1)T , and v3=(0,1,2)T ,...Ch. 3.5 - Given v1=(12) , v2=(23) , S=(351 2) find vectors...Ch. 3.5 - Given v1=(26) , v2=(14) , S=(4121) find vectors u1...Ch. 3.5 - Let [x,1] and [2x1,2x+1] beorderedbasesfor P2 ....Ch. 3.5 - Find the transition matrix representing the...Ch. 3.5 - Let E={u1,...,un} and F={v1,...,vn} be two ordered...Ch. 3.6 - For each of the following matrices, find a basis...Ch. 3.6 - In each of the following, determine the dimension...Ch. 3.6 - Let A=(122314245549367859) (a) Compute the reduced...Ch. 3.6 - For each of the following choices of A and b,...Ch. 3.6 - For each consistent system in Exercise 4,...Ch. 3.6 - How many solutions will the linear system Ax=b...Ch. 3.6 - Let A be a 6n matrix of rank r and let b be a...Ch. 3.6 - Let Abe an mn matrix with mn . Let bRm and suppose...Ch. 3.6 - Let A and B be 65 matrices. If dimN(A)=2 ,what is...Ch. 3.6 - Let A be an mn matrix whose rank is equal to n. If...Ch. 3.6 - Let A be an mn matrix. Prove that rank(A)min(m,n)Ch. 3.6 - Let A and B be row equivalent matrices. (a) Show...Ch. 3.6 - Let A be a 43 matrixandsupposethatthevectors...Ch. 3.6 - Let A be a 44 matrix with reduced row echelonform...Ch. 3.6 - Let A be a 45 matrix and let U be the reduced row...Ch. 3.6 - Let A be a 58 matrix with rank equal to 5 and let...Ch. 3.6 - LetA bea 45 matrix, If a1,a2 , and a4 are...Ch. 3.6 - Let A be a 53 matrix of rank 3 and let {x1,x2,x3}...Ch. 3.6 - Let A be an mnmatrixwithrankequalton.Showthat if...Ch. 3.6 - Prove that a linear system Ax=b is consistent...Ch. 3.6 - LetAandBbemn matrices.Showthat...Ch. 3.6 - Let Abeanmn matrix. (a) Show that if B is a...Ch. 3.6 - Prove Corollary 3.6.4.Ch. 3.6 - Show that if A and B are nn matrices and N(AB)=n...Ch. 3.6 - Prob. 25ECh. 3.6 - Prob. 26ECh. 3.6 - Let x and y be nonzero vectors in m and n ,...Ch. 3.6 - Prob. 28ECh. 3.6 - Let Amn , Bnr , and C=AB . Show that (a) ifA and B...Ch. 3.6 - Prob. 30ECh. 3.6 - An mn matrix A is said to have a right inverse if...Ch. 3.6 - Prove: If A is an mn matrix and the column vectors...Ch. 3.6 - Show that a matrix B has a left inverse if and...Ch. 3.6 - Let B be an nm matrix whose columns arelinearly...Ch. 3.6 - Prob. 35ECh. 3.6 - Prob. 36ECh. 3 - (Change of Basis) Set U=round(20rand(4))10 ,...Ch. 3 - (RankDeficient Matrices) In this exercise we...Ch. 3 - (Column Space arid Reduced Row Echelon Form) Set...Ch. 3 - (Rank1 Updates of Linear Systems) (a) Set...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Prob. 5CTACh. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Prob. 10CTACh. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Answer each of the statements that follows as true...Ch. 3 - Prob. 14CTACh. 3 - Prob. 15CTACh. 3 - In 3 , let x1 and x2 be linearly independent...Ch. 3 - For each set that follows determine whether it is...Ch. 3 - Let A=(13134001110022200333) (a) Find a basis for...Ch. 3 - How do the dimensions of the null space and column...Ch. 3 - Answer the following questions and, in each case,...Ch. 3 - Let S be the set of all symmetric 22 matrices with...Ch. 3 - Let A be a 64 matrix of rank 4. (a) What is the...Ch. 3 - Given the vectors x1=(122),x2=(133) ,...Ch. 3 - Let x1,x2 and x3 be linearly independent vectors...Ch. 3 - Let A be a 65 matrix with linearly independent...Ch. 3 - Let {u1,u2} and {v1,v2} be ordered bases for 2 ,...
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