1. Consider the traffic flow problem. We have a traffic network and for each intersection we...
1. Consider the traffic flow problem. We have a traffic network and for each intersection we have measured how many cars are leaving the network there (negative numbers for cars entering).
(a) Explain the linear system of equations we need to solve. It should involve the incidence matrix of the network (or its transpose) and some right hand side vector.
(b) Supposed the system has no solution. Because of measuring errors, the number of cars entering the network is not equal to the number of cars leaving the network. We find the least squares solution instead. Explain what is the error that this least squares solution minimizes.
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1. Consider the traffic flow problem. We have a traffic network
and for each intersection we...
Solve in maple if possible. 5. (15 points) One application of linear systems of equations is...
Solve in maple if possible.
5. (15 points) One application of linear systems of equations is traffic analysis. Below is a diagram of part of the Inner Harbor area of Baltimore. The problem you want to explore is the amount of traffic among the eight intersections (labeled A, B, H). The arrows indicate the direction of traffic. Each street is a one-way street. The numbers indicate the number of vehicles (per hour) that enters or leaves an intersection. For example,...
Matrix Algebra Consider the traffic flow diagram that follows, where a1, az, a3, 24, 61, 62,...
Matrix Algebra
Consider the traffic flow diagram that follows, where a1, az, a3, 24, 61, 62, 63, 64 are fixed positive integers. Set up a linear system in the unknowns X1, X2, X3, X4 and show that the system will be consistent if and only if a1 + a2 + a3 + 24 = 61 + b2 + b3 + 64 What can you conclude about the number of automobiles entering and leaving the traffic network? ai ba bi X1...
3. (25 pts) Consider the data points: t y 0 1.20 1 1.16 2 2.34 3 6.08 ake a least squares fitting of these data using the model yü)- Be + Be-. Suppose we want to m (a) Explain how you would comput...
3. (25 pts) Consider the data points: t y 0 1.20 1 1.16 2 2.34 3 6.08 ake a least squares fitting of these data using the model yü)- Be + Be-. Suppose we want to m (a) Explain how you would compute the parameters β | 1 . Namely, if β is the least squares solution of the system Χβ y, what are the matrix X and the right-hand side vector y? what quantity does such β minimize? (b)...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions. If A is square and invertible, there is a unique solution, which can be written as x = A-'b. The concept of pseudoinverse seeks to generalise this idea to non-square matrices and to cases (ii) and (iii). Taking case (ii) of an inconsistent linear system, we may solve the normal equations AT Ar = Ab...
Problem 2. In this problem we consider the question of whether a small value of the...
Problem 2. In this problem we consider the question of whether a small value of the residual kAz − bk means that z is a good approximation to the solution x of the linear system Ax = b. We showed in class that, kx − zk kxk ≤ kAkkA −1 k kAz − bk kbk . which implies that if the condition number kAkkA−1k of A is small, a small relative residual implies a small relative error in the solution....
1. Consider the three points in the zy plane (see left). We want to find a...
1. Consider the three points in the zy plane (see left). We want to find a line y = mx +n that goes through all three points, if possible. (4,5) (a) If there is such a line, the following three equations must be true. The first equation is given. Write the remaining two equations. (2,3) (1,2) 2 = 1 m +n (1) (b) Rewrite (1) in Av = b format. v= | is the unknown vector for which we want...
Problem 2 We have seen that, if a matrix A is invertible, then we can express...
Problem 2 We have seen that, if a matrix A is invertible, then we can express the unique solution of Az = b as I = A-15. Soon, we will introduce ideas that help us understand At = b better when A is not invertible. This problem is preparation for that! 11 301 1-3 -9 2 Let A = 2 6 0 1-2 -6 -5 (a) Solve the system A7 = 5. (b) What does the solution set of Az...
Answer ALL the questions. Some or all of them shall be marked. Question 1. Consider the...
Answer ALL the questions. Some or all of them shall be marked. Question 1. Consider the following system of differential equations: P.(D) [x] + P (D)) -(0) Px(D) [x] + P (D) x = f(t). (1) How do we determine the correct number of arbitrary constants in a general solution of the above system. (0) Explain briefly the difference between the operator method and the method of triangu- larization when used for solving the above system. Question 2. Determine whether...
1. Consider a dilute solution of molecules at fixed temperature T. These molecules have access to...
1. Consider a dilute solution of molecules at fixed temperature T. These molecules have access to a surface that has a total of B binding sites where molecules can bind. To count states in this system, we will divide space into small cells that each can hold a single molecule. There are a total of B cells that have a binding site, and a total of M cells that do not have binding sites. The overall number of cells is...
Problem 4.1 - Odd Bound States for the Finite Square Well Consider the finite square well...
Problem 4.1 - Odd Bound States for the Finite Square Well Consider the finite square well potential of depth Vo, V(x) = -{ S-V., –a sx sa 10, else In lecture we explored the even bound state solutions for this potential. In this problem you will explore the odd bound state solutions. Consider an energy E < 0 and define the (real, positive) quantities k and k as 2m E K= 2m(E + V) h2 h2 In lecture we wrote...
Solve in maple if possible.
5. (15 points) One application of linear systems of equations is traffic analysis. Below is a diagram of part of the Inner Harbor area of Baltimore. The problem you want to explore is the amount of traffic among the eight intersections (labeled A, B, H). The arrows indicate the direction of traffic. Each street is a one-way street. The numbers indicate the number of vehicles (per hour) that enters or leaves an intersection. For example,...
Matrix Algebra
Consider the traffic flow diagram that follows, where a1, az, a3, 24, 61, 62, 63, 64 are fixed positive integers. Set up a linear system in the unknowns X1, X2, X3, X4 and show that the system will be consistent if and only if a1 + a2 + a3 + 24 = 61 + b2 + b3 + 64 What can you conclude about the number of automobiles entering and leaving the traffic network? ai ba bi X1...
3. (25 pts) Consider the data points: t y 0 1.20 1 1.16 2 2.34 3 6.08 ake a least squares fitting of these data using the model yü)- Be + Be-. Suppose we want to m (a) Explain how you would compute the parameters β | 1 . Namely, if β is the least squares solution of the system Χβ y, what are the matrix X and the right-hand side vector y? what quantity does such β minimize? (b)...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions. If A is square and invertible, there is a unique solution, which can be written as x = A-'b. The concept of pseudoinverse seeks to generalise this idea to non-square matrices and to cases (ii) and (iii). Taking case (ii) of an inconsistent linear system, we may solve the normal equations AT Ar = Ab...
1. Consider the three points in the zy plane (see left). We want to find a line y = mx +n that goes through all three points, if possible. (4,5) (a) If there is such a line, the following three equations must be true. The first equation is given. Write the remaining two equations. (2,3) (1,2) 2 = 1 m +n (1) (b) Rewrite (1) in Av = b format. v= | is the unknown vector for which we want...
Problem 2 We have seen that, if a matrix A is invertible, then we can express the unique solution of Az = b as I = A-15. Soon, we will introduce ideas that help us understand At = b better when A is not invertible. This problem is preparation for that! 11 301 1-3 -9 2 Let A = 2 6 0 1-2 -6 -5 (a) Solve the system A7 = 5. (b) What does the solution set of Az...
Answer ALL the questions. Some or all of them shall be marked. Question 1. Consider the following system of differential equations: P.(D) [x] + P (D)) -(0) Px(D) [x] + P (D) x = f(t). (1) How do we determine the correct number of arbitrary constants in a general solution of the above system. (0) Explain briefly the difference between the operator method and the method of triangu- larization when used for solving the above system. Question 2. Determine whether...
1. Consider a dilute solution of molecules at fixed temperature T. These molecules have access to a surface that has a total of B binding sites where molecules can bind. To count states in this system, we will divide space into small cells that each can hold a single molecule. There are a total of B cells that have a binding site, and a total of M cells that do not have binding sites. The overall number of cells is...
Problem 4.1 - Odd Bound States for the Finite Square Well Consider the finite square well potential of depth Vo, V(x) = -{ S-V., –a sx sa 10, else In lecture we explored the even bound state solutions for this potential. In this problem you will explore the odd bound state solutions. Consider an energy E < 0 and define the (real, positive) quantities k and k as 2m E K= 2m(E + V) h2 h2 In lecture we wrote...
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