Canonical - Canonical Forms of Linear Systems Lecture IV I....

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1 Canonical Forms of Linear Systems Lecture IV I. Systems of Equations with the Same Solution A. Consistent equations and linear combinations 12 31 32 2: 2 7: xx xE -+= +-= 1. The ordered set x 1 =3, x 2 =2, x 3 =1 is said to be a solution of E 1 because the values substituted into E 1 yield 2=2. Thus, (3,2,1) is said to satisfy equation E 1 . In general, for an arbitrary mxn (m equations and n unknowns) the solution (x 1 1 ,x 2 1 , …x n 1 ) is a solution of the i th equation if 1 11 1 1 22 i i i n ni a x a x a xb + += L 2. There are three possibilities for a given system. a. The system possesses a unique solution. b. The system possesses an infinite number of solutions. c. The system possesses no solution. 3. A system that possesses solutions whether unique or not is called consistent or solvable while a system containing no solution is inconsistent or unsolvable. 4. The aggregate of solutions of a system is called the solution set. If the system is inconsistent its solution set is said to be empty (or a null set). a. Given a system it is easy to construct new equations such that any solution of the original equations also solves the new equation. ( 29 ( 29 123 122 3 27 2224 63 3 21 45 5 17 xxx - --+ =- Notice that (3,2,1) solves the original system 3212 2(3 )217 That point also solves the new system 4(3 ) 5(2 ) 5(1 ) 1 2 1 05 --+= -- +=
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AEB 6592 Canonical Forms Professor Charles B. Moss 2 5. A scheme for generating new equations is then ( 29 ( 29 ( 29 1 1 1 1 1 2 211 2 2 1 1 2 2 222 1 1 22 1 1 1 1 1 1 2 2 11 2 1 1 2 2 2 1 1 1 1 nn m m m m n nm mm nnn m mn ka x a x a xb k a x a x a d x d x d xd d k a k a d k a k a d k a k a d k b k b kb + += + + + =++ L L M L L L L M L L a. An equation formed in this manner is called a linear combination of the original equations. The numbers i k are called multipliers or weights of the linear combination. b. Writing these in detached form [ ] 1 1 1 2 1 2 1 2 2 2 12 n n m m m n n a a a bk a a a a a a d ddd    L L M MO M MM L L 6. Definition: If in a system of equations an equation is a linear combination of the other equations, it is said to be dependent upon them; the dependent equation is called redundant. A vacuous equation, i.e. an equation of the form 0 0 00 n x xx + L is also called redundant when it occurs in a single equation system. A system containing no redundancy is called independent ( 29 ( 29 123 1 23 1 5 304 3 5 1 5 20 3 6 1 6 xxx x xx x ++= ++-= + -= + Therefore, in the system of equations 1 1 34 3 6 1 6 x + + the last equation is redundant B. How systems are solved 1. The usual “elimination” procedure for finding a solution of a system of
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Canonical - Canonical Forms of Linear Systems Lecture IV I....

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