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HOMOGENEOUS LINEAR SYSTEMS

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

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HOMOGENEOUS LINEAR SYSTEMS

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  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: • Long-hand: • Vector form: where

  3. or more visually explicit: Finally • 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 • The system is consistent (it has at least a solution.) • 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

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