Exercise 2.2.14. Create a 2 by 2 system of equations that has many homogeneous solutions but no particular solution.

Answer: A trivial example of such a system is where and ; for example

The corresponding homogeneous system

has any vector in as a solution, but the general system has no particular solution since for any .

For a less trivial example we can have both rows of equal to each other, but have the corresponding elements of not be equal to each other. For example, consider the system

The corresponding homogeneous system

has infinitely many solutions of the form

but there is no particular solution to the general system: Elimination produces the following system

and the second row produces the contradictory equation .

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.