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 again expressing the final result to three significant digits:

Note that the values of are almost identical, though the values of the solutions are very different. This indicates that the solution to is very sensitive to small differences in .

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.