homogeneous linear systems

1. HOMOGENEOUS LINEAR SYSTEMS
Up to now we have been studying linear systems
of the form
We intend to make life easier for ourselves by
choosing the vector
to be the zero-vector

2. We write the new, easier equation in the three
familiar equivalent forms:
1. Long-hand:
2. Vector form:
where

3. or more visually explicit:
Finally
3. The concise form

4. In any of the three forms, a linear system with an
augmented matrix having zeroes in the rightmost
column is called a
homogeneous linear system.
Homogeneous linear systems have very nice
solution sets
before proceeding with our study we need to
establish a couple of useful facts about the
product
Fact 1.
Fact 2.

5. We can already say something nice about the
solution set of
(From Fact 1) If a vector
(From Fact 2) If two vectors
are solutions, then so is their sum
This says that the solution set S of a homoge-
neous linear system is kind of
once you are in it you can’t get out using either

6. In there are few distinct kinds of sets that are
lines through the origin
planes through the origin and
In fact, the origin is the one guaranteed solution of
a homogeneous linear system
It makes sense to ask the question
Are there any non-zero (aka non-trivial) solutions?
Let’s return to the echelon form of the matrix
We know that

7. (p.43 of the textbook)
This statement will allow us to describe precisely the
solution set of An example will show how.

8. Let be the matrix shown below (we are in )
We find the solutions of
using the row-reduction
program downloaded from
the class website. The
reduced echelon form is
We get the two
equations

9. In vector form the solution is:
In other words, the solution set consists of
all scalar multiples of
If instead of we write we can say:
Solution set

10. Let’s do another example. Here is a matrix
Let’s find all the solutions
of the homogeneous lin-
ear system
Using our program we
obtain that the reduced echelon form of
We get the equations

11. In vector form we get
. We get

12. that tells us that the solution set is …
a plane in
Note how the two vectors
are read off from
Can you formulate a
rule? Careful, think of

13. The textbook calls the equality
The Parametric Vector Form of the solution set.
What about the old (non-homogeneous) friend
We will take care of it next.
There are obviously two cases
1 The system is consistent (it has at least a
solution.)
2 The system is inconsistent (no solutions.)
We know when 2 happens, the rightmost column of
the augmented matrix has a pivot term.
What can we say about 1 ?

14. Let’s begin by naming
To say that the linear system
is consistent is to say that
pick one
On the other hand, we just finished describing
In details. We assert:
Our statement can be proved as a fairly simple
Corollary of the following

15. rather powerful
Theorem. Let
The proof of the theorem comes directly from the
two properties of we have studied before.
Note that the theorem describes precisely all the
solutions of