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

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

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

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