Exercise 2.1.8. Consider the following system of linear equations:

Does the set of solutions form a point, line, or plane? Is it a subspace? Is it the nullspace of ? The column space of ?

Answer: The system corresponds to

From the second equation we have and we can substitute into the first equation to obtain or . We therefore have so that the set of solutions can be represented as

where is a free variable. The set of solutions is therefore a line passing through the origin and the point .

Since the solution set is a line passing through the origin it is a subspace, and since is the set of all vectors for which it is by definition the nullspace of (see page 68).

The column space of is the set of all vectors that are linear combinations of the columns of :

and thus contains 2 by 1 vectors. All solution vectors are 3 by 1 vectors; the set of solutions is not the same as the column space of .

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.