HOMOGENEOUS LINEAR SYSTEMSUp 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

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

2. Vector form:

where

or more visually explicit:

Finally

3. The concise form

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 setsbefore proceeding with our study we need to establish a couple of useful facts about the productFact 1.Fact 2.

We can already say something nice about the solution set of (From Fact 1) If a vector

(From Fact 2) If two vectorsare solutions, then so is their sumThis says that the solution set S of a homoge-neous linear system is kind ofonce you are in it you can’t get out using either

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

(p.43 of the textbook)

This statement will allow us to describe precisely the solution set of An example will show how.

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

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

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

In vector form we get

. We get

that tells us that the solution set is … a plane in

Note how the two vectorsare read off from

Can you formulate a

rule? Careful, think of

The textbook calls the equalityThe 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 cases1 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 ?

Let’s begin by naming

To say that the linear systemis consistent is to say that pick oneOn the other hand, we just finished describingIn details. We assert: Our statement can be proved as a fairly simple Corollary of the following

rather powerfulTheorem. 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