1.5 Solution Sets of Linear Systems

1
1.5 Solution Sets of Linear Systems
2
Definition Homogeneous
A system of linear equations is said to be
homogeneous if it can be written in the form
where A is an matrix and
is the zero vector in .
Example
Note Every homogeneous linear system is
consistent.
i.e. The homogeneous system has at
least one solution, namely the trivial solution,
.
3
Important Question

• When does a homogenous system have a
  non-trivial solution?
• That is, when does it have a non-zero vector
  such that ?

4
Example1 Determine if the following homogeneous
system has a nontrivial solution
Geometrically, what does the solution set
represent?
5
Basic variables the variables corresponding to
pivot columns
Free variables the others
The homogeneous equation has a
nontrivial solution if and only if the equation
has at least one free variable.
6
Example2 Describe all solutions of the
homogeneous system
Geometrically, what does the solution set
represent?
7
Solutions of Nonhomogeneous Systems
Example2 Describe all solutions for
i.e. Describe all solutions of
where
and
Geometrically, what does the solution set
represent?
8
Homogeneous
Nonhomogeneous
9
Homogeneous
Nonhomogeneous
y
y
x
x
z
z
10
Theorem Suppose is consistent for
some given , and let be a solution.
Then the solution set of is the
set of all vectors of the form
where is any solution of the
homogeneous equation .