Exercise 3.1.16. Describe the set of all vectors orthogonal to the vectors and .

Answer: If a vector is orthogonal to the vectors and then its inner products with those vectors must be zero, so that

and

This is a system of two equations in four unknowns:

and is equivalent to the system where

In other words, the problem of finding all vectors orthogonal to and is equivalent to the problem of finding the nullspace of the 2 by 4 matrix for which and are the rows.

We use Gaussian elimination to solve the system, starting by subtracting 2 times row 1 from row 2:

The resulting echelon matrix has two pivots, in columns 1 and 2, so and are basic variables and and are free variables.

Setting and , from row 2 we have and from row 1 we then have or . So is one solution to . Setting and , from row 2 we have again and from row 1 we have or . So is a second solution to .

The vectors and together form a basis for the nullspace of . That nullspace, i.e., all linear combinations of and , is the set of all vectors orthogonal to the vectors and .

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.