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UNIT17

# 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 Ax = b has either no solution, or exactly one solution, or or infinitely 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. Ay = b and Az = b with z = y for t let x = (1 t ) y + t ( z ) (there are infinitely many such x ’s since t is arbitrary). Then Ax = A ((1 t ) y + tz ) = A ( y + t ( z y )) = Ay + t ( Az Ay ) = b + t ( b b ) = b + 0 = b 1

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Therefore, if we have more than one solution, we have infinitely many solutions. Definition 18.2. The linear system Ax = b is said to be homogeneous if b = 0. If b = 0 then the system is said to be non-homogeneous . Example 1. Identify the following systems as homogeneous or non-homogeneous. (a) 1 1 2 0 0 1 2 0 6 3 2 7 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 = 0 (c) x 1 2 x 2 + 3 x 3 = 0 x 1 2 x 3 = 0 non-homogeneous system. 5 x 2 + x 3 = 1 Definition 18.3. The zero vector 0 is a solution to every homogeneous sys- tem Ax = 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 Ax = 0 has either ex- actly one solution or infinitely many. Proof: We showed that for any linear system there were only 3 options; no solution, one solution, or infinitely many solutions, and any homogeneous
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UNIT17 - Unit 17 The Theory of Linear Systems Theorem 18.1...

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