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.

Hi hecker, while we doing the A’ 3X3 matrix, since the information that x1+2×2+4×3=0 is given, can we be sure that A’ matrix is a rank1 matrix??

A’ has rank 1 because the second and third columns are multiples of the first column, and thus are linearly dependent on the first column. The rank of a matrix is the number of linearly independent columns, and in this case there is only 1 such column.