homogeneous linear systems up to now we have been studying linear systems of the form we intend to...
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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
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
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