Toán học - Chapter 3: Systems of linear equations

Math Dept, Faculty of Applied Science, HCM University of Technology ------------------------------------------------------------------------------------- Chapter 3: Systems of Linear Equations • Instructor Dr. Dang Van Vinh (11/2006) CONTENTS --------------------------------------------------------------------------------------------------------------------------- I – Systems of Linear Equations II – Homogeneous system I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- The quantities a11, a12, , amn are called the coefficients of the system. 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 n n n n m m mn m m a x a x a x b a x a x a x b a x a x a x b                               A System of m linear equations in n unknowns has the form: Definition of a System of Linear equations. The quantities b1, b2, , bm are called the free or constant terms of the systems. I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- The solution of system is such a collection n of the numbers c1, c2, , cm which, being substituted into system for the unknowns x1, x2, , xn, turns all the equations of the system into identities. A system is called homogeneous if all its constant terms b1, b2, , bm are equal to zero. Definition of Homogeneous System. A system is called nonhomogeneous if at least one of the constant terms b1, b2, , bm is different from zero. Definition of Nonhomogeneous System. I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- consistent system inconsistent system A system of linear equations has either: 1. no solution, 2. exactly one solution, or 3. infinitely many solutions Two linear systems are equivalent, if they have the same solution set. The basic strategy is to replace one system with an equivalent system that is easier to solve. I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- There are three elementary reduction operations . An operation is called elementary reduction operation if it transforms one system to equivalent system. Definition of elementary reduction operation 1. Swapping: an equation is swapped with another. 2. Rescaling (multiplying by a scalar): an equation has both sides multiplied by a nonzero constant. 3. Pivoting: an equation is replaced by the sum of itself and a multiple of another. Remark: We need to prove that above three operations are elementary reduction operations. I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- 1 2 1 3 2r r r r      0 3 3 3 3 3 x y y z y z           2 3 r r  0 3 3 3 4 0 x y y z z          Solution: x = 1; y = -1; z = 0 Solve the system: 0 2 3 3 2 3 x y x y z x y z           Example I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- 1 1 0 2 1 3 1 2 1          coefficient matrix: augmented matrix: 1 1 0 0 2 1 3 3 1 2 1 3           I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- 1 2 1 3 2r r r r      2 3 r r  1 1 0 0 2 1 3 3 1 2 1 3           1 1 0 0 0 3 3 3 0 3 1 3           1 1 0 0 0 3 3 3 0 0 4 0          I. System of Linear Equations The variable corresponding to pivot columns in the matrix are called basic variable. The other variable is called free variable. Definition of Basic and Free variable. 1 1 1 2 1 2 2 3 5 6 3 3 4 1 1          row operations 1 1 1 2 1 0 0 1 1 4 0 0 0 6 8           x1, x3, x4: basic variables x2: free variables If , then the system AX = b is consistent. ( | ) ( )r A b r A If , then the system AX = b is inconsistent. ( | ) ( )r A b r A I. System of Linear Equations If = number of unknowns , then the system AX = b has unique solution. ( | ) ( )r A b r A If < number of unknowns , then the system AX = b has many solutions. ( | ) ( )r A b r A The Kronecker Capelli Theorem If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. I. System of Linear equations -------------------------------------------------------------------------------------------------------- --- 1. Write the augmented matrix of the system. 2. Use the row reduction algorithm to obtain an equivalent augmented matrix is echelon form. Decide whether the system is consistent. 3. Write the system of equations corresponding to the matrix 4. Rewrite each nonzero equation from step 3 so that its one basic variable is expressed in terms of any free variable appearing in the equation. Using Row Reduction to solve a linear system I. Systems of Linear Equations ---------------------------------------------------------------------------------------------------------------------- The augmented matrix of a linear system has been transformed by a row operations into the form below. Determine if the system is consistent. 1 5 2 6 . 0 4 7 2 , 0 0 5 0 a          1 1 1 3 . 0 1 2 4 , 0 0 0 5 b          1 1 1 0 . 0 1 2 5 , 0 0 0 0 c          1 1 1 0 . 0 3 1 0 . 0 0 0 0 c          Example I. Systems of Linear Equations -------------------------------------------------------------------------------------------------------------------- Example 5 2 1 4 6 3 3 9 x y z x y z x y z              Solve the following system: I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- 3 3 5 9 2 2 3 3 y z x y z x y z            Example Solve the following system: I. Systems of Linear Equation --------------------------------------------------------------------------------------------------------------------------- Basis variable: 521 ,, xxx Free variable: 43, xx General Solution: 1 2 3 4 5 24 2 3 7 2 2 4 x x x x x                      Find the general solution of the linear system Example 2 3 4 5 1 2 3 4 5 1 2 3 4 5 3 6 6 4 5 3 7 8 5 8 9 3 9 12 9 6 15 x x x x x x x x x x x x x x                  I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- Find the general solutions of the systems whose augmented matrix is given as below Example 1 1 1 1 2 3 4 1 3 4 2 1          I. Systems of Linear Equations ----------------------------------------------------------------------------------------------------------- --- Find the general solutions of the systems whose augmented matrix is given as below Example 1 1 2 0 2 1 5 0 3 4 5 0          I. Systems of Linear Equations ----------------------------------------------------------------------------------------------------------- -- Find the general solutions of the systems whose augmented matrix is given as below Example 1 1 1 1 2 2 1 3 0 1 3 4 2 2 5 2 3 1 1 3            I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- Find the general solutions of the systems whose augmented matrix is given as below 1 1 2 0 1 2 3 1 2 4 3 4 5 1 3 1 2 3 1 0             Example I. Systems of Linear Equations ------------------------------------------------------------------------------------------------------------ - Determine the value(s) m such that the matrix is the augmented of a consistent linear system Example 2 1 1 1 1 1 , 1 1 m m m m m          I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- 1 1 1 1 2 3 1 4 3 4 1m m          Determine the value(s) m such that the matrix is the augmented of a consistent linear system Example I. Systems of Linear Equations ---------------------------------------------------------------------------------------------------------- Example Determine the value(s) m such that the corresponding linear system has unique solution 1 1 1 1 1 2 1 3 1 2 , 3 4 2 0 6 2 1 0 1m m              I. Systems of Linear Equations --------------------------------------------------------------------------------------------------------------------------- Example Determine the value(s) m such that the corresponding linear system has unique solution 2 2 3 1 4 0 3 2 1 5 7 1 1 1m m          II. Homogeneous Systems. --------------------------------------------------------------------------------------------------------------------------- A system is called homogeneous if all its constant terms b1, b2, , bm are equal to zero. Definition of Homogeneous System. The homogeneous system always has zero solution x1 = x2 = = xn = 0. This solution is called the trivial solution. The homogeneous system always possesses a unique solution – the trivial solution - if and only if rank (A) = n. II. Homogeneous Systems. --------------------------------------------------------------------------------------------------------------------------- The homogeneous system AX = 0 has nontrivial solution if and only if rank (A) < n. The homogeneous system AX = 0, where A is a square matrix, has nontrivial solution if and only if det(A) = 0. II. Homogeneous Systems. --------------------------------------------------------------------------------------------------------------------------- Determine the general solution for the following homogeneous system. Example 1 2 3 4 1 2 3 4 1 2 3 4 2 2 0 2 4 3 0 3 6 4 0 x x x x x x x x x x x x               II. Homogeneous Systems. --------------------------------------------------------------------------------------------------------------------------- Among all solutions that satisfy the homogeneous system Example 2 0 2 4 0 2 0 x y z x y z x y z            Determine those also satisfy the nonlinear constraint y – xy = 2z II. Homogeneous Systems. --------------------------------------------------------------------------------------------------------------------------- If A is the coefficient matrix for a homogeneous system consisting of four equations in eight unknowns and if there are five free variables, what is rank (A)? Example Explain why a homogeneous system of m equations in n unknowns where m < n must always possess an infinite number of solutions. Example I. Systems of Linear Equations ------------------------------------------------------------------------------------------------------------ - Determine the value(s) m such that the homogeneous system has nontrivial solution Example 0 2 3 5 0 3 ( 1) 0 x y z x y z x my m z            