Exercise 2.4.10. Suppose the nullspace of a matrix is the set of all vectors in for which . Find a 1 by 3 matrix with this nullspace. Find a 3 by 3 matrix with the same nullspace.
Answer: If and then we have . We know that so one way to construct a suitable matrix is to set , , and so that .
For a 3 by 3 matrix , if each row of times is 0 so that for . Since we can construct a suitable matrix by setting each row of to or to any multiple of that vector. For example, one possible choice of is
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.