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Tutorial10 - Systems of first-order linear equations: I liu...

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Systems of first-order linear equations: I liu dongwen email: ldxab@ust.hk Before introducing the general theory of systems of first order linear equations, let us make a review of linear algebra. Properties of matrices. An m × n matrix A consists of a rectangular ar- ray of numbers, arranged in m rows and n columns, that is A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . a m 1 a m 2 ··· a mn . The notation ( a ij ) is used to denote the matrix whose generic element is a ij . Let A = ( a ij ) , B = ( b ij ) , C = ( c ij ) in the sequel. 1. Transpose. A T = ( a ji ) . 2. conjugate. A = ( a ij ) . 3. Addition and subtraction. If A and B are both m × n , then A ± B = ( a ij ± b ij ) . 4. Multiplication by a number. Let λ be a scalar, then λ A = ( λa ij ) . 5. Multiplication. If A is m × n and B is n × r , then C = AB is an m × r matrix, defined by c ij = n X k =1 a ik b kj . In special, if A = a b c d , x = x 1 x 2 , then Ax = ax 1 + bx 2 cx 1 + dx 2 . In general, we have ( AB ) T = B T A T , AB = ( A )( B ) . Note that in general AB 6 = BA . 6. Identity. The multiplicative identity I , is given by I = 1 0 ··· 0 0 1 ··· 0 . . . . . . . . . 0 0 ··· 1 . We have AI = IA = A for any (square) matrix A . 7. Inverse and determinant. A square matrix A is invertible if and only if det A 6 = 0 . In this case there is another matrix B such that AB = BA = I . This B is usually denoted by A - 1 . For a 2 × 2 matrix A = a b c d , 1
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one has det A = ad - bc. In general, det AB = (det A )(det B ) . If A is n × n and λ is a scalar, then det( λ A ) = λ n det A . Systems of linear algebraic equations. A set of n simultaneous linear algebraic equations in n variables, a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = b 1 , . . . a n 1 x 1 + a n 2 x 2 + ··· + a nn x n = b n , can be written as Ax = b , (1) where the n × n matrix A and the vector b are given, and the components of x are to be determined. The system is said to be homogeneous if b = 0 ; otherwise it is nonhomogeneous . To solve a linear system like (1), we shall transform the augmented matrix
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Tutorial10 - Systems of first-order linear equations: I liu...

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