UNIT17

# UNIT17 - Unit 17 The Theory of Linear Systems Theorem 18.1....

This preview shows pages 1–3. Sign up to view the full content.

Unit 17 The Theory of Linear Systems Theorem 18.1. Every system of linear equations A~x = ~ b has either no solution, or exactly one solution, or or inFnitely many solutions (i.e. parametric family of solutions) Proof: We have seen examples of each kind, so we need only show that there are no more possibilities. Suppose ~y and ~z are two distinct solutions i.e. A~y = ~ b and A~z = ~ b with ~z 6 = ~y for t ∈< let ~x =(1 t ) ~y + t ( ~z ) (there are inFnitely many such ~x ’s since t is arbitrary). Then A~x = A ((1 t ) ~y + t~z ) = A ( ~y + t ( ~z ~y )) = A~y + t ( A~z A~y ) = ~ b + t ( ~ b ~ b ) = ~ b + ~ 0= ~ b 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Therefore, if we have more than one solution, we have inFnitely many solutions. Defnition 18.2. The linear system A~x = ~ b is said to be homogeneous if ~ b = ~ 0. If ~ b 6 = ~ 0 then the system is said to be non-homogeneous . Example 1. Identify the following systems as homogeneous or non-homogeneous. (a) 1 120 0120 6327 x y z = 0 0 0 homogeneous system. (b) 3 x 1 +2 x 2 + x 3 =0 3 x 2 x 3 = 0 homogeneous system. 2 x 1 + x 3 (c) x 1 2 x 2 +3 x 3 x 1 2 x 3 = 0 non-homogeneous system. 5 x 2 + x 3 =1 Defnition 18.3. The zero vector ~ 0 is a solution to every homogeneous sys- tem A~x = ~ 0, since A ~ 0= ~ 0. The solution ~x = 0 is called the trivial solution , any other solution is called a nontrivial solution . Theorem 18.4. Every homogeneous linear system A~x = ~ 0 has either ex- actly one solution or inFnitely many. ProoF: We showed that for any linear system there were only 3 options; no solution, one solution, or inFnitely many solutions, and any homogeneous
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 04/15/2010 for the course MATH 20C taught by Professor Lit during the Spring '10 term at UCLA.

### Page1 / 6

UNIT17 - Unit 17 The Theory of Linear Systems Theorem 18.1....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online