Tag Archives: least squares

Linear Algebra and Its Applications, Exercise 3.3.18

Exercise 3.3.18.¬†Suppose we have¬†the following measurements of : and want to fit a plane of the form . a) Write a system of 4 equations in 3 unknowns representing the problem. (The system may not have a solution.) b) Write … Continue reading

Posted in linear algebra | Tagged , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.3.13

Exercise 3.3.13. Using least squares, find the line that is the best fit to the following measurements: at at at at Also, given the matrix find the projection of onto the column space . Answer: Assuming that the line in … Continue reading

Posted in linear algebra | Tagged , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.3.5

Exercise 3.3.5. Given the system with no solution, provide a graph of a straight line that minimizes and solve for the equation of the line. What is the result of projecting the vector onto the column space of ? Answer: … Continue reading

Posted in linear algebra | Tagged , , , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.3.4

Exercise 3.3.4. Given expand the expression , compute its partial derivatives with respect to and , and set them to zero. Compare the resulting equations to to confirm that you obtain the same normal equations in both cases (i.e., using … Continue reading

Posted in linear algebra | Tagged , , , | 2 Comments

Linear Algebra and Its Applications, Exercise 3.3.3

Exercise 3.3.3. Given solve to find . Show that is orthogonal to every column in . Answer: We have or if is invertible. In this case we have so that We then have and The error vector is then The … Continue reading

Posted in linear algebra | Tagged , , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.3.2

Exercise 3.3.2. We have the value at time and the value at time . We wish to fit these values using a line constrained to go through the origin, i.e., with an equation of the form . Solve the system … Continue reading

Posted in linear algebra | Tagged , | Leave a comment

Linear Algebra and Its Applications, Exercise 3.3.1

Exercise 3.3.1. a) Given the system of equations consisting of and find the least squares solution and describe the error being minimized by that solution. Confirm that the error vector is orthogonal to the column . Answer: This is a … Continue reading

Posted in linear algebra | Tagged , | Leave a comment