Exercise 3.1.9. For the plane in 
spanned by the vectors 
and 
find the orthogonal complement (i.e., the line in perpendicular to the plane). Note that this can be done by solving the system 
where the two vectors are the rows of 
.
Answer: We have
We solve the system using Gaussian elimination, starting by subtracting 1 times row 1 from row 2:
Since the resulting echelon matrix has pivots in columns 1 and 2, we have 
and 
as basic variables and 
as a free variable. Setting 
, from the second row we have 
or 
. From the first row we have 
or 
.
So 
is a solution to and a basis for the orthogonal complement, which consists of the line through the origin 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
.
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