Category Archives: linear algebra

Exercise 1.7.9. Given the matrix compare the pivots in standard elimination with those used in partial pivoting. Answer: In the standard elimination process we would use a multiplier (1/.001) to multiply the first row and subtract it from the second: … Continue reading →

Exercise 1.7.8. Take the 10 by 10 Hilbert matrix for which and solve the equation . Make a minor change to or and see how the solution changes. Answer: We can use the open source statistics software R for this … Continue reading →

Exercise 1.7.7. Take the 3 by 3 Hilbert matrix from the previous exercise and compute assuming that and are solutions to . Answer: We first multiply times : expressing the final result to three significant digits. We then multiply by … Continue reading →

Exercise 1.7.6. Given the 3 by 3 Hilbert matrix find its inverse (i) using an exact calculation and (ii) rounding off all values to three digits. Answer: (i) We first multiply the first row times and subtract it from the … Continue reading →

Exercise 1.7.5. What would the difference matrix in equation (6) look like if the boundary conditions were and (instead of and )? Answer: The finite difference equation would still be as given in equation (5): For this equation would become … Continue reading →

Exercise 1.7.4. For the differential equation with the corresponding difference equation is for . Solve the above equation for and compare their values to the true solution at ,  and . Answer: We do Gaussian elimination on the difference matrix and … Continue reading →

Exercise 1.7.3. Given the differential equation what is the finite-difference matrix for (corresponding to a 5 by 5 matrix)? Note that you can replace the boundary condition by (and hence ) and the boundary condition by (and hence ). Show … Continue reading →

Exercise 1.7.2. Given the differential equation what is the finite difference matrix for (corresponding to a 3 by 3 matrix)? Answer: We have so that the equation above can be expressed in finite-difference terms as replacing continuous values of by … Continue reading →

Exercise 1.7.1. Find the LDU factorization for the matrix (This is the same matrix as in equation (6) of section 1.7, only with instead of .) Answer: For the first step of elimination we multiply the first row times the … Continue reading →

Exercise 1.6.23. Assume that and are square matrices, and that is invertible. Show that is invertible as well. (Use the fact that .) Answer: First, since and are square matrices we know that both of the product matrices and exist … Continue reading →