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 1.4, Problem 34E
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You are provided with three 2D data points, p1, p2 and p3. Solving A C = B for C provides youwith the coefficients of a natural cubic spline curve that interpolates these points.Additionally, you have been given A and B, but some elements are missing. Moreover, the last two rowsof A are entirely absent. Your task is to determine and fill in the missing elements. For the last two rows,enforce a zero tangent at the beginning (in p1) and a not-a-knot boundary condition in p2. The matricesA and B are given as follows:Explain how to find the entries of A and B . How would you adapt these matrices if the data pointswere 3D? What if your spline should go through five data points? How many “extra rows” would there thenbe (with “extra” meaning “in addition to securing C2-continuity”)?
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Chapter 1 Solutions
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Ch. 1.1 - Use back substitution to solve each of the...Ch. 1.1 - Write out the coefficient matrix for each of the...Ch. 1.1 - In each of the following systems, interpret each...Ch. 1.1 - Write an augmented matrix for each of the systems...Ch. 1.1 - Write out the system of equations that corresponds...Ch. 1.1 - Solve each of the following systems. (a)...Ch. 1.1 - The two systems 2x1+x2=34x1+3x2=5 and...Ch. 1.1 - Solve the two systems...Ch. 1.1 - Given a System of the form m1x1+x2=b1m2x1+x2=b2...Ch. 1.1 - Consider a system of the form...
Ch. 1.1 - Give a geometrical interpretation of a linear...Ch. 1.2 - Which of the matrices that follow are in row...Ch. 1.2 - The augmented matrices that follow are in row...Ch. 1.2 - The augmented matrices that follow are in reduced...Ch. 1.2 - For each of the systems in Exercise 3, make a list...Ch. 1.2 - For each of the systems of equations that follow,...Ch. 1.2 - Use GaussJordan reduction to solve each of the...Ch. 1.2 - Give a geometric explanation of why a homogeneous...Ch. 1.2 - Consider a linear system whose augmented matrix is...Ch. 1.2 - Consider a linear system whose augmented matrix is...Ch. 1.2 - Consider a linear system whose augmented matrix is...Ch. 1.2 - Given the linear systems...Ch. 1.2 - Given the linear systems (i)...Ch. 1.2 - Given a homogeneous system of linear equations, if...Ch. 1.2 - Given a nonhomogeneous system of linear equations,...Ch. 1.2 - Determine the values ofx1,x2,x3,x4for the...Ch. 1.2 - Prob. 16ECh. 1.2 - Prob. 17ECh. 1.2 - In Application 3 the solution (6, 6, 6, 1) was...Ch. 1.2 - Prob. 19ECh. 1.2 - Nitric acid is prepared commercially by a series...Ch. 1.2 - Prob. 21ECh. 1.2 - Prob. 22ECh. 1.3 - If A=[314201122]andB=[102311241] compute (a) 2A...Ch. 1.3 - For each of the pairs of matrices that follow,...Ch. 1.3 - For which of the pairs in Exercise 2 is it...Ch. 1.3 - Write each of the following systems of equations...Ch. 1.3 - If A=[341127] verify that (a) 5A=3A+2A (b)...Ch. 1.3 - If A=[ 4 2 1 3 6 5 ]andB=[ 1 2 3 2 2 4 ] verify...Ch. 1.3 - If A=[216324]andB=[2416] verify that (a)...Ch. 1.3 - If A=[ 2 1 4 3 ],B=[ 2 0 1 4 ],C=[ 3 2 1 1 ]...Ch. 1.3 - Let A=[ 1 1 2 2 ],b=[40],c=[32] (a) Write b as a...Ch. 1.3 - For each of the choices of A and b that follow,...Ch. 1.3 - Let Abe a 53 matrix. If b=a1+a2=a2+a3 then what...Ch. 1.3 - Let Abe a 34 matrix. If b=a1+a2+a3+a4 then what...Ch. 1.3 - Let Ax=b be a linear system whose augmented matrix...Ch. 1.3 - Prob. 14ECh. 1.3 - Let A be an mn matrix. Explain why the matrix...Ch. 1.3 - A matrix A is said to be skew symmetric if AT=A ....Ch. 1.3 - In Application 3, suppose that we are searching...Ch. 1.3 - Let A be a 22 matrix with a110 and let =a21/a11 ....Ch. 1.4 - Explain why each of the following algebraic rules...Ch. 1.4 - Will the rules in Exercise 1 work if a is replaced...Ch. 1.4 - Find nonzero 22 matrices A and B such that AB=0 .Ch. 1.4 - Find nonzero matrices A, B, and C such that...Ch. 1.4 - The matrix A=[1111] has the property that A2=O ....Ch. 1.4 - Prove the associative law of multiplication for 22...Ch. 1.4 - Let A=[ 1 2 1 2 1 2 1 2] Compute A2 and A3 . What...Ch. 1.4 - Let A=[ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1...Ch. 1.4 - Let A=[0100001000010000] Show that An=O for n4 .Ch. 1.4 - Let A and B be symmetric nn matrices. For each of...Ch. 1.4 - Let C be nonsymmetric nn matrix. For each of the...Ch. 1.4 - Let A=[ a 11 a 12 a 21 a 22] Show that if...Ch. 1.4 - Use the result from Exercise 12 to find the...Ch. 1.4 - Let A and B are nn matrices. Show that if...Ch. 1.4 - Let A be a nonsingular matrix. Show that A1 is...Ch. 1.4 - Prove that if A is nonsingular then AT is...Ch. 1.4 - Let A be an nn matrix and let x and y be vectors...Ch. 1.4 - Let A be a nonsingular nn matrix. Use mathematical...Ch. 1.4 - Let A be an nn matrix. Show that if A2=O , then IA...Ch. 1.4 - Let A be an nn matrix. Show that if Ak+1=O , then...Ch. 1.4 - Given R=[cossinsincos] Show that R is nonsingular...Ch. 1.4 - An nn matrix A is said to be an involutionifA2=I ....Ch. 1.4 - Let u be a unity vector in n (i.e. uTu=1 ) and let...Ch. 1.4 - A matrix A is said to be an idempotentif A2=A ....Ch. 1.4 - Prob. 25ECh. 1.4 - Let D be an nn diagonal matrix whose diagonal...Ch. 1.4 - Let Abe an involution matrix and let...Ch. 1.4 - Let A be an mn matrix. Show that ATA and AAT are...Ch. 1.4 - Let A and B be symmetric nn matrices. Prove that...Ch. 1.4 - Let Abe an nn matrix and let B=A+ATandC=AAT (a)...Ch. 1.4 - In Application 1, how many married women and how...Ch. 1.4 - Consider the matrix A=[ 0 1 0 1 1 1 0 1 1 0 0 1 0...Ch. 1.4 - Consider the graph (a) Determine the adjacency...Ch. 1.4 - If Ax=Bx for some nonzero vector x, then the...Ch. 1.4 - If A and B are singular nn matrices, then A+B is...Ch. 1.4 - If A and B are nonsingular matrices, then (AB)T is...Ch. 1.5 - Which of the matrices that follow are elementary...Ch. 1.5 - Find the inverse of each matrix in Exercise 1. For...Ch. 1.5 - Prob. 3ECh. 1.5 - Prob. 4ECh. 1.5 - Let A=[ 1 2 1 2 1 0 4 3 2 ],B=[ 1 2 2 2 1 2 4 3 6...Ch. 1.5 - Prob. 6ECh. 1.5 - Let A=[2164] (a) Express A1 as a product of...Ch. 1.5 - Compute the LU factorization of each of the...Ch. 1.5 - Let A=[ 1 3 2 0 3 2 1 4 3 ] (a) Verify that A1=[ 1...Ch. 1.5 - Find the inverse of each of the following...Ch. 1.5 - Prob. 11ECh. 1.5 - Let A=[ 5 3 3 2 ],B=[ 6 2 2 4 ],C=[ 4 6 2 3 ]...Ch. 1.5 - Is the transpose of an elementary matrix an...Ch. 1.5 - Let U and R bennupper triangular matrices and...Ch. 1.5 - Let A be a 33 matrix and suppose that 2a1+a24a3=0...Ch. 1.5 - Prob. 16ECh. 1.5 - Let A and B be nn matrices and let C=AB . Show...Ch. 1.5 - Prob. 18ECh. 1.5 - Let U be an nn upper triangular matrix with...Ch. 1.5 - Prob. 20ECh. 1.5 - Prob. 21ECh. 1.5 - Show that if A is a symmetric nonsingular matrix...Ch. 1.5 - Prove that if A is a row equivalent to B then B is...Ch. 1.5 - Prob. 24ECh. 1.5 - Prob. 25ECh. 1.5 - Prove that B is row equivalent to A if and only if...Ch. 1.5 - Is it possible for a singular matrix B to be row...Ch. 1.5 - Prob. 28ECh. 1.5 - Prob. 29ECh. 1.5 - Prob. 30ECh. 1.5 - Prob. 31ECh. 1.5 - Prob. 32ECh. 1.6 - Let A be a nonsingular nn matrix. Perform the...Ch. 1.6 - Prob. 2ECh. 1.6 - Let A=[ 1 2 1 1 ]andB=[ 2 1 1 3 ] (a) Calculate...Ch. 1.6 - Let I=[ 1 0 0 1 ],E=[ 0 1 1 0 ],O=[ 0 0 0 0 ] C=[...Ch. 1.6 - Perform each of the following block...Ch. 1.6 - Given X=[ 2 4 1 2 5 3 ],Y=[ 1 2 2 3 4 1 ] (a)...Ch. 1.6 - Let A=[ A 11 A 21 A 12 A 22 ]andAT=[ A 11 T A 12 T...Ch. 1.6 - Let Abe an mn matrix, X and nr matrix, and B an mn...Ch. 1.6 - Prob. 9ECh. 1.6 - Prob. 10ECh. 1.6 - Prob. 11ECh. 1.6 - Let A and B be nn matrices and let M be a block...Ch. 1.6 - Prob. 13ECh. 1.6 - Prob. 14ECh. 1.6 - Prob. 15ECh. 1.6 - Prob. 16ECh. 1.6 - Prob. 17ECh. 1.6 - Let A, B, L, M, S, and T be nn matrices with A, B,...Ch. 1.6 - Let Abe an nn matrix and xn . (a) A scalar c can...Ch. 1.6 - If A is an nn matrix with the property that Ax=0...Ch. 1.6 - Prob. 21ECh. 1.6 - Prob. 22ECh. 1 - Use MATLAB to generate random 44 matrices A and B....Ch. 1 - Set n=200 and generate an nn matrix and two...Ch. 1 - Set A=floor(10*rand(6)) . By construction, the...Ch. 1 - Construct a mainx as follows: Set...Ch. 1 - Generate a matrix A by setting A = floor(10 *...Ch. 1 - Consider the graph (a) Determine the adjacency...Ch. 1 - In Application 1 of Section 1.4, the numbers of...Ch. 1 - The following table describes a seven-stage model...Ch. 1 - Set A = magic(8) and then compute its reduced row...Ch. 1 - Set B=[1,1;1,1] and A=[zeros(2),eye(2);eye(2),B]...Ch. 1 - The MATLAB commands A = floor(10 * rand((6)), B=AA...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - Find all solutions of the linear system...Ch. 1 - (a) A linear equation in two unknowns corresponds...Ch. 1 - LetAx=bbe a system of n linear equations in n...Ch. 1 - LetAbeamatrix of the form A=[22] where and are...Ch. 1 - Let A=[213427135],B=[213135427],C=[013027535] Find...Ch. 1 - Let A be a 33 matrix and let b=3a1+a2+4a3 Will the...Ch. 1 - Let A be a 33 matrix and suppose that a13a2+2a3=0...Ch. 1 - Given the vector x0=[11] is it possible to find 22...Ch. 1 - Let A and B be symmetric nn matrices and let C=AB...Ch. 1 - Let E and F be nn elementary matrices and let C=EF...Ch. 1 - Given A=[IOOOIOOBI] where all the submatrices are...Ch. 1 - LetA and B be 1010 matrices that are...
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