Exercise 3.1.12. For the matrix
find a basis for the nullspace and show that it is orthogonal to the row space. Take the vector 
and express it as the sum of a nullspace component 
and a row space component 
.
Answer: We use Gaussian elimination to solve the system 
. We start by subtracting 1 times row 1 from row 2:
The resulting echelon matrix has two pivots, in columns 1 and 2, so 
and 
are basic variables and 
is a free variable. Setting 
from the second row we have 
or 
. From the first row we then have 
or 
. The vector 
is therefore a solution to and a basis for the nullspace.
Since is in the nullspace so is its negative, 
, which is an alternative basis for the nullspace. Note that the inner product of 
with 
is
and that the inner product of with 
is
The basis of the nullspace is thus orthogonal to each of the basis vectors for the row space, and . The row space as a whole consists of all linear combinations 
of those basis vectors, where 
and 
are scalars.
We then have
so that the basis of the nullspace is orthogonal to all vectors in the row space.
Note that we have
Since the vector 
is in the nullspace and the vector is in the row space (being the second row of 
) we have 
where 
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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