Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
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Chapter 1.1, Problem 46E
Boris and Marina are shopping for chocolate bars. Borisobserves, “If I add half my money to yours, it willbe enough to buy Iwo chocolate bars.’ Marina naivelyasks, “If I add half my money to yours, how manycan we buy?” Boris replies, “One chocolate bar.” Howmuch money did Boris have? (From Yuri Chernyak and Robert Rose, The Chicken from Minsk, Basic Books,1995.)
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You are given a plane Π in R3 defined by two vectors, p1 and p2, and a subspace W in R3 spanned by twovectors, w1 and w2. Your task is to project the plane Π onto the subspace W.First, answer the question of what the projection matrix is that projects onto the subspace W and how toapply it to find the desired projection. Second, approach the task in a different way by using the Gram-Schmidtmethod to find an orthonormal basis for subspace W, before then using the resulting basis vectors for theprojection. Last, compare the results obtained from both methods
Chapter 1 Solutions
Linear Algebra with Applications (2-Download)
Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...Ch. 1.1 - GOAL Set up and solve systems with as many as...
Ch. 1.1 - In Exercises 11 through 13,find all solutions of...Ch. 1.1 - In Exercises 11 through 13, find all solutions of...Ch. 1.1 - In Exercises 11 through 13, find all solutions of...Ch. 1.1 - In Exercises 14 through 16,find all solutions of...Ch. 1.1 - In Exercises 14 through 16, find all solutions of...Ch. 1.1 - In Exercises 14 through 16, find all solutions of...Ch. 1.1 - Find all solutions of the linear system | x+2y=a...Ch. 1.1 - Find all solutions of the linear system...Ch. 1.1 - Consider the linear system...Ch. 1.1 - Consider the linear system |x+yz=2x+2y+z=3x+y+( k...Ch. 1.1 - The sums of any two of three real numbers are 24,...Ch. 1.1 - Emile and Gertrude are brother and sister. Emile...Ch. 1.1 - Consider a two-commodity market. When the...Ch. 1.1 - The Russian-born U.S. economist and Nobel laureate...Ch. 1.1 - Find the outputs a andb needed to satisfy the...Ch. 1.1 - Consider the differential equation...Ch. 1.1 - Find all solutions of the system |7xy=x6x+8y=y| ,...Ch. 1.1 - On a sunny summer day, you are taking the...Ch. 1.1 - On your next trip to Switzerland, you should take...Ch. 1.1 - In a grid of wires, the temperature at exterior...Ch. 1.1 - Find the polynomial of degree 2 [a polynomial of...Ch. 1.1 - Find a polynomial of degree 2 [of the form...Ch. 1.1 - Find all the polynomials f(t) of degree 2 [of the...Ch. 1.1 - Find all the polynomials f(t) of degree 2 [of the...Ch. 1.1 - Find all the polynomials f(t) of degree 2 [of the...Ch. 1.1 - Find all the polynomials f(t) of degree 2 [of the...Ch. 1.1 - Find the function f(t) of the form f(t)=ae3t+be2t...Ch. 1.1 - Find the function f(t) of the form...Ch. 1.1 - Prob. 39ECh. 1.1 - Find the ellipse centered at the origin that runs...Ch. 1.1 - Find all points (a,b,c) in space for which the...Ch. 1.1 - Linear systems are particularly easy to solve when...Ch. 1.1 - Consider the linear system |x+y=1x+ t 2y=t| ,...Ch. 1.1 - Find a system of linear equations with three...Ch. 1.1 - Find a system of linear equations with three...Ch. 1.1 - Boris and Marina are shopping for chocolate bars....Ch. 1.1 - Here is another method to solve a system of linear...Ch. 1.1 - A hermit eats only two kinds of food: brown rice...Ch. 1.1 - I have 32 bills in my wallet, in the denominations...Ch. 1.1 - Some parking meters in Milan, Italy, accept coins...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - GOAL Use Gauss-Jordan elimination to solve linear...Ch. 1.2 - Solve the linear systems in Exercises 13 through...Ch. 1.2 - Solve the linear systems in Exercises 13 through...Ch. 1.2 - Solve the linear systems in Exercises 13 through...Ch. 1.2 - Prob. 16ECh. 1.2 - Solve the linear systems in Exercises 13 through...Ch. 1.2 - Determine which of the matrices below are in...Ch. 1.2 - Find all 41 matrices in reduced row-echelon form.Ch. 1.2 - For which values of a, b, c, d, and e is the...Ch. 1.2 - For which values of a, b, c, d, and e is the...Ch. 1.2 - We say that two nm matrices in reduced...Ch. 1.2 - How many types of 32 matrices in reduced...Ch. 1.2 - How many types of 23 matrices in reduced...Ch. 1.2 - Prob. 25ECh. 1.2 - Suppose matrix A is transformed into matrix B...Ch. 1.2 - Prob. 27ECh. 1.2 - Consider an nm in matrix A. Can you transform...Ch. 1.2 - Prob. 29ECh. 1.2 - Suppose you subtract a multiple of an equation in...Ch. 1.2 - Balancing a chemical reaction. Consider the...Ch. 1.2 - Find the polynomial of degree 3 [a polynomial of...Ch. 1.2 - Find the polynomial of degree 4 whose graph...Ch. 1.2 - Cubic splines. Suppose you are in charge of the...Ch. 1.2 - Find the polynomial f(t) of degree 3 such that...Ch. 1.2 - The dot product of two vectors x=[ x 1 x 2 x n]...Ch. 1.2 - Find all vectors in 4 that are perpendicular to...Ch. 1.2 - Find all solutions x1,x2,x3 of the equation...Ch. 1.2 - Prob. 39ECh. 1.2 - If we consider more than three industries in an...Ch. 1.2 - Consider the economy of Israel in 1958.11 The...Ch. 1.2 - Prob. 42ECh. 1.2 - Prob. 43ECh. 1.2 - The accompanying sketch represents a maze of...Ch. 1.2 - Let S(t) be the length of the tth day of the year...Ch. 1.2 - Prob. 46ECh. 1.2 - Consider the equations...Ch. 1.2 - Consider the equations |y+2kz=0x+2y+6z=2kx+2z=1| ,...Ch. 1.2 - a. Find all solutions x1,x2,x3,x4 of the system...Ch. 1.2 - For an arbitrary positive integer n3 , find all...Ch. 1.2 - Prob. 51ECh. 1.2 - Find all the polynomials f(t) of degree 3 such...Ch. 1.2 - Prob. 53ECh. 1.2 - Prob. 54ECh. 1.2 - Prob. 55ECh. 1.2 - Prob. 56ECh. 1.2 - Prob. 57ECh. 1.2 - Prob. 58ECh. 1.2 - Prob. 59ECh. 1.2 - Prob. 60ECh. 1.2 - Prob. 61ECh. 1.2 - Prob. 62ECh. 1.2 - Students are buying books for the new semester....Ch. 1.2 - Prob. 64ECh. 1.2 - At the beginning of a political science class at a...Ch. 1.2 - Prob. 66ECh. 1.2 - Prob. 67ECh. 1.2 - Prob. 68ECh. 1.2 - Prob. 69ECh. 1.2 - Prob. 70ECh. 1.2 - Prob. 71ECh. 1.2 - Prob. 72ECh. 1.2 - Pigeons are sold at the rate of 5 for 3 panas,...Ch. 1.2 - Prob. 74ECh. 1.2 - Prob. 75ECh. 1.2 - Prob. 76ECh. 1.2 - Prob. 77ECh. 1.2 - Prob. 78ECh. 1.2 - Prob. 79ECh. 1.2 - Prob. 80ECh. 1.3 - GOAL Use the reduced row-echelon form of the...Ch. 1.3 - Find the rank of the matrices in Exercises 2...Ch. 1.3 - Find the rank of the matrices in Exercises 2...Ch. 1.3 - Find the rank of the matrices in Exercises 2...Ch. 1.3 - a. Write the system |x+2y=73x+y=11| in vector...Ch. 1.3 - Consider the vectors v1,v2,v3 in 2 (sketched in...Ch. 1.3 - Consider the vectors v1,v2,v3 in 2 shown in the...Ch. 1.3 - Consider the vectors v1,v2,v3,v4 in 2 shown in...Ch. 1.3 - Write the system |x+2y+3z=14x+5y+6z=47x+8y+9z=9|...Ch. 1.3 - Compute the dot products in Exercises 10 through...Ch. 1.3 - Compute the dot products in Exercises 10 through...Ch. 1.3 - Compute the dot products in Exercises 10 through...Ch. 1.3 - Compute the products Axin Exercises 13 through 15...Ch. 1.3 - Compute the products Axin Exercises 13 through 15...Ch. 1.3 - Compute the products Axin Exercises 13 through 15...Ch. 1.3 - Compute the products Axin Exercises 16 through 19...Ch. 1.3 - Compute the products Axin Exercises 16 through 19...Ch. 1.3 - Compute the products Axin Exercises 16 through 19...Ch. 1.3 - Compute the products Axin Exercises 16 through 19...Ch. 1.3 - a. Find [234567]+[753101] . b. Find 9[112345] .Ch. 1.3 - Use technology to compute the product...Ch. 1.3 - Consider a linear system of three equations with...Ch. 1.3 - Consider a linear system of four equations with...Ch. 1.3 - Let A be a 44 matrix, and let b and c be two...Ch. 1.3 - Let A be a 44 matrix, and let b and c be two...Ch. 1.3 - Let A be a 43 matrix, and let b and c be two...Ch. 1.3 - If the rank of a 44 matrix A is 4, what is...Ch. 1.3 - If the rank of a 53 matrix A is 3, what is...Ch. 1.3 - In Problems 29 through 32, let x=[539]andy=[201]....Ch. 1.3 - In Problems 29 through 32, let x=[539]andy=[201]....Ch. 1.3 - In Problems 29 through 32, let x=[539]andy=[201]....Ch. 1.3 - In Problems 29 through 32, let x=[539]andy=[201]....Ch. 1.3 - Let A be the nn matrix with all 1‘s on the...Ch. 1.3 - We define the vectors e1=[001],e2=[010],e3=[001]...Ch. 1.3 - In m , we define ei=[0010]ithcomponent . If A is...Ch. 1.3 - Find a 33 matrix A such that...Ch. 1.3 - Find all vectors x such that Ax=b , where...Ch. 1.3 - Prob. 38ECh. 1.3 - Prob. 39ECh. 1.3 - Prob. 40ECh. 1.3 - Prob. 41ECh. 1.3 - Prob. 42ECh. 1.3 - Prob. 43ECh. 1.3 - Consider an nm matrix A with more rows than...Ch. 1.3 - Prob. 45ECh. 1.3 - Prob. 46ECh. 1.3 - A linear system of the form Ax=0 is called...Ch. 1.3 - Consider a solution x1 of the linear system Ax=b...Ch. 1.3 - Consider the accompanying table. For some linear...Ch. 1.3 - Consider a linear system Ax=b , where A is a 43...Ch. 1.3 - Consider an nm matrix A, an rs matrix B, and...Ch. 1.3 - Consider the matrices A=[1012] and B=[0110] .Can...Ch. 1.3 - If A and B are two nm matrices, is (A+B)x=Ax+Bx...Ch. 1.3 - Prob. 54ECh. 1.3 - Prob. 55ECh. 1.3 - Is the vector [301385662] a linear combination of...Ch. 1.3 - Prob. 57ECh. 1.3 - For which values of the constants b and c is the...Ch. 1.3 - For which values of the constants c and d is...Ch. 1.3 - For which values of the constants a, b, c and d is...Ch. 1.3 - For which values of the constant c is [1cc2] a...Ch. 1.3 - For which values of the constant c is [1cc2] a...Ch. 1.3 - In Exercises 63 through 68, consider the vectors...Ch. 1.3 - In Exercises 63 through 68, consider the vectors...Ch. 1.3 - Prob. 65ECh. 1.3 - Prob. 66ECh. 1.3 - Prob. 67ECh. 1.3 - Prob. 68ECh. 1.3 - Prob. 69ECh. 1.3 - Let A be the nn matrix with 0’s on the main...Ch. 1 - TRUE OR FALSE? 19 Determine whether the statements...Ch. 1 - TRUE OR FALSE? 19 Determine whether the statements...Ch. 1 - Matrix [120001000] is in reduced row-echelon form.Ch. 1 - A system of four linear equations in three...Ch. 1 - There exists a 34 matrix with rank 4.Ch. 1 - If A is a 34 matrix and vector v is in 4 , then...Ch. 1 - If the 44 matrix A has rank 4, then any linear...Ch. 1 - There exists a system of three linear equations...Ch. 1 - There exists a 55 matrix A of rank 4 such that the...Ch. 1 - If matrix A is in reduced row-echelon form, then...Ch. 1 - The system [123456000]x=[123] is inconsistent.Ch. 1 - There exists 22 matrix A such that A=[12]=[34] .Ch. 1 - If A is a nonzero matrix of the form [abba] , then...Ch. 1 - rank [111123136]=3Ch. 1 - The system Ax=[0001] is inconsistent for all 43...Ch. 1 - There exists a 22 matrix A such that A=[11]=[12]...Ch. 1 - rank [222222222]=2Ch. 1 - [111315171921][131]=[131921]Ch. 1 - There exists a matrix A such that A=[12]=[357] .Ch. 1 - Vector [123] is a linear combination of vectors...Ch. 1 - If the system Ax=b has a unique solution, then...Ch. 1 - If A is any 43 matrix, then there exists a vector...Ch. 1 - There exist scalars a and b such that matrix...Ch. 1 - If v and w are vectors in 4 , then v must be a...Ch. 1 - If u,v , and w are nonzero vectors in 2 , then w...Ch. 1 - If v and w are vectors in 4 , then the zero vector...Ch. 1 - If A and B are any two 33 matrices of rank2,then...Ch. 1 - If vector u is a linear combination of vectors v...Ch. 1 - A linear system with fewer unknowns than...Ch. 1 - The rank of any upper triangular matrix is the...Ch. 1 - There exists a 43 matrix A of rank 3 such that...Ch. 1 - The system Ax=b is inconsistent if (and only...Ch. 1 - If A is a 43 matrix of rank 3 and Au=Aw for two...Ch. 1 - If A is a 44 matrix and the system Ax=[2345] has...Ch. 1 - If vector u is a linear combination of vectors v...Ch. 1 - If A=[uvw] and rref(A)=[002013000] , then the...Ch. 1 - If A and B are matrices of the same size, then the...Ch. 1 - If A and B are any two nn matrices of rank n, then...Ch. 1 - If a vector v in 4 is a linear combination of u...Ch. 1 - If matrix E is in reduced row-echelon form, and if...Ch. 1 - The linear system Ax=b consistent if (and only if)...Ch. 1 - If A is a 34 matrix of rank 3, then the system...Ch. 1 - If two matrices A and B have the same reduced...Ch. 1 - If matrix E is in reduced row-echelon form, and if...Ch. 1 - If A and B are two 22 matrices such that the...Ch. 1 - A lower triangular 33 matrix has rank 3 if (and...Ch. 1 - If adbc0 , then the matrix [abcd] must have rank...Ch. 1 - If vector w is a linear combination of u and v ,...Ch. 1 - If the linear system Ax=b has a unique solution...Ch. 1 - A matrix is called a 0-1-matrix if all of its...
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