Solution sets of nonhomogeneous systems of linear
equations
Theorem. Let p be one solution to the nonhomogeneous equation and let S
be the set of
all solutions to the associated homogeneous equation
Ax = 0.
Then the solution set of Ax = b consists of all vectors in the set p + S.
Example. Consider the linear system Ax = b whose
augmented matrix is
Linear independence
Example. Let
We have already seen that span
is the plane P in R3 given by the equation
What happens to the span if we add a third vector
to
the set of vectors generating the
span? In other words, what is span 
?
Definition. A (linear) dependence relation among a set of
vectors 
is an
equation of the form
where 
for some vector
.
Example.
Definition. If there exists a dependence relation
among a set of vectors ,
then we say that the set is linearly dependent. A
set is linearly independent if it is not linearly dependent.
Matrix characterization
A dependence relation 
can be rewritten as
the matrix equation
Ar = 0 where
Therefore, a dependence relation among the vectors
is the same as a nontrivial
solution to Ar = 0.
Example. Consider the vectors
We can de termine that these three vectors are linearly
independent by considering the matrix
Example. Which of the fol lowing sets of vectors in R3 are
linearly independent? Why?
