SOLUTION SETS for LINEAR SYSTEMS
 Homogeneous Systems
 homogeneous system
 solution types trivial
,
otherwise nontrivial

has a nontrivial solution iff there is
at least one free variable
 solution is in parametric form when it is written as
a linear combination of vectors
 Nonhomogeneous Systems
 nonhomogeneous system
 complete solution in terms of particular and homogeneous solutions:
if
and
then
is also a solution.
 geometry: if the general solution is
,
this is the
line through
parallel to .
 Writing a Solution Set
 1.
 Row reduce the augmented matrix
 2.
 Express the basic variables in terms of free variables
 3.
 Write
in terms of constants and free variables
 4.
 Write
as a linear combination of vectors (parametric form)
LINEAR INDEPENDENCE
 Independence
 The set
is
linearly independent if
has only the trivial solution.
 The set
is
linearly dependent if there exist
,
not all
zero, such that
 Matrix Column Independence

The columns of A are linearly independent iff
has only
the trivial solution.
 Characterization

are linearly
dependent if some
is a linear combination of the other
vectors.
 Simple Checks for Dependence

are linearly dependent
if k > n.

is linearly dependent if
.
Alan C Genz
20000706