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Page 1 Linear System ? Linear equation: ax 1 + bx 2 = c where, a, b, c = constants x 1, x 2 = variables ? A linear system consists of more than one linear equations. ? For example: a 1 x 1 + b 1 x 2 + c 1 x 3 = k 1 ……………….(1) a 2 x 1 + b 2 x 2 + c 2 x 2 = k 2 ……………….(2) a 3 x 1 + b 3 x 2 + c 3 x 3 = k 3 ……………….(3) ? To find a solution of this system means to find the value/values of x 1, x 2 , x 3 which satisfies all the equations in the linear system. ? A linear system can have a unique solution, more than one solutions or no solution. Page 2 Linear System ? Linear equation: ax 1 + bx 2 = c where, a, b, c = constants x 1, x 2 = variables ? A linear system consists of more than one linear equations. ? For example: a 1 x 1 + b 1 x 2 + c 1 x 3 = k 1 ……………….(1) a 2 x 1 + b 2 x 2 + c 2 x 2 = k 2 ……………….(2) a 3 x 1 + b 3 x 2 + c 3 x 3 = k 3 ……………….(3) ? To find a solution of this system means to find the value/values of x 1, x 2 , x 3 which satisfies all the equations in the linear system. ? A linear system can have a unique solution, more than one solutions or no solution. Linear System ? Rewriting the above equation in form of matrix AX = B A = ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 ?? 3 ?? 3 ?? 3 , X = ?? 1 ?? 2 ?? 3 , B = ?? 1 ?? 2 ?? 3 ? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented matrix of the linear system. Type eq uati on h er e. Augmented matrix : ?? 1 ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 ?? 2 ?? 3 ?? 3 ?? 3 ?? 3 ? If all the elements of B are zero then linear system is called Homogeneous otherwise Non Homogeneous. Page 3 Linear System ? Linear equation: ax 1 + bx 2 = c where, a, b, c = constants x 1, x 2 = variables ? A linear system consists of more than one linear equations. ? For example: a 1 x 1 + b 1 x 2 + c 1 x 3 = k 1 ……………….(1) a 2 x 1 + b 2 x 2 + c 2 x 2 = k 2 ……………….(2) a 3 x 1 + b 3 x 2 + c 3 x 3 = k 3 ……………….(3) ? To find a solution of this system means to find the value/values of x 1, x 2 , x 3 which satisfies all the equations in the linear system. ? A linear system can have a unique solution, more than one solutions or no solution. Linear System ? Rewriting the above equation in form of matrix AX = B A = ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 ?? 3 ?? 3 ?? 3 , X = ?? 1 ?? 2 ?? 3 , B = ?? 1 ?? 2 ?? 3 ? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented matrix of the linear system. Type eq uati on h er e. Augmented matrix : ?? 1 ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 ?? 2 ?? 3 ?? 3 ?? 3 ?? 3 ? If all the elements of B are zero then linear system is called Homogeneous otherwise Non Homogeneous. Row Reduced Echelon Form of a Matrix ? A matrix C is said to be in the row reduced form if 1. The first nonzero entry in each row of C is 1 2. The column containing this 1 has all its other entries zero e.g. A = 1 1 3 0 3 2 0 0 1 B = 1 0 1  2 0 1 2 3 0 0 4 1 0 0  1  2 C = 1  1 0 2 Page 4 Linear System ? Linear equation: ax 1 + bx 2 = c where, a, b, c = constants x 1, x 2 = variables ? A linear system consists of more than one linear equations. ? For example: a 1 x 1 + b 1 x 2 + c 1 x 3 = k 1 ……………….(1) a 2 x 1 + b 2 x 2 + c 2 x 2 = k 2 ……………….(2) a 3 x 1 + b 3 x 2 + c 3 x 3 = k 3 ……………….(3) ? To find a solution of this system means to find the value/values of x 1, x 2 , x 3 which satisfies all the equations in the linear system. ? A linear system can have a unique solution, more than one solutions or no solution. Linear System ? Rewriting the above equation in form of matrix AX = B A = ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 ?? 3 ?? 3 ?? 3 , X = ?? 1 ?? 2 ?? 3 , B = ?? 1 ?? 2 ?? 3 ? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented matrix of the linear system. Type eq uati on h er e. Augmented matrix : ?? 1 ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 ?? 2 ?? 3 ?? 3 ?? 3 ?? 3 ? If all the elements of B are zero then linear system is called Homogeneous otherwise Non Homogeneous. Row Reduced Echelon Form of a Matrix ? A matrix C is said to be in the row reduced form if 1. The first nonzero entry in each row of C is 1 2. The column containing this 1 has all its other entries zero e.g. A = 1 1 3 0 3 2 0 0 1 B = 1 0 1  2 0 1 2 3 0 0 4 1 0 0  1  2 C = 1  1 0 2 Gauss Elimination Method ? Gaussian elimination is a method of solving a linear system AX = B (consisting of m equations in n unknowns) by bringing the augmented matrix [A B] = ?? 11 ??1 2 ? ??1 ?? ??1 ??2 1 ??2 2 ? ??2 ?? ??2 ? ? ? ? ? ????1 ????2 ? ?????? ???? ? to an upper triangular form (or reduced row echelon form) ?? 11 ?? 12 ? ?? 1?? ?? 1 0 ?? 22 ? ?? 2?? ?? 2 ? ? ? ? ? 0 0 ? ?? ???? ?? ?? ? This elimination process is also called the forward elimination method. Page 5 Linear System ? Linear equation: ax 1 + bx 2 = c where, a, b, c = constants x 1, x 2 = variables ? A linear system consists of more than one linear equations. ? For example: a 1 x 1 + b 1 x 2 + c 1 x 3 = k 1 ……………….(1) a 2 x 1 + b 2 x 2 + c 2 x 2 = k 2 ……………….(2) a 3 x 1 + b 3 x 2 + c 3 x 3 = k 3 ……………….(3) ? To find a solution of this system means to find the value/values of x 1, x 2 , x 3 which satisfies all the equations in the linear system. ? A linear system can have a unique solution, more than one solutions or no solution. Linear System ? Rewriting the above equation in form of matrix AX = B A = ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 ?? 3 ?? 3 ?? 3 , X = ?? 1 ?? 2 ?? 3 , B = ?? 1 ?? 2 ?? 3 ? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented matrix of the linear system. Type eq uati on h er e. Augmented matrix : ?? 1 ?? 1 ?? 1 ?? 1 ?? 2 ?? 2 ?? 2 ?? 2 ?? 3 ?? 3 ?? 3 ?? 3 ? If all the elements of B are zero then linear system is called Homogeneous otherwise Non Homogeneous. Row Reduced Echelon Form of a Matrix ? A matrix C is said to be in the row reduced form if 1. The first nonzero entry in each row of C is 1 2. The column containing this 1 has all its other entries zero e.g. A = 1 1 3 0 3 2 0 0 1 B = 1 0 1  2 0 1 2 3 0 0 4 1 0 0  1  2 C = 1  1 0 2 Gauss Elimination Method ? Gaussian elimination is a method of solving a linear system AX = B (consisting of m equations in n unknowns) by bringing the augmented matrix [A B] = ?? 11 ??1 2 ? ??1 ?? ??1 ??2 1 ??2 2 ? ??2 ?? ??2 ? ? ? ? ? ????1 ????2 ? ?????? ???? ? to an upper triangular form (or reduced row echelon form) ?? 11 ?? 12 ? ?? 1?? ?? 1 0 ?? 22 ? ?? 2?? ?? 2 ? ? ? ? ? 0 0 ? ?? ???? ?? ?? ? This elimination process is also called the forward elimination method. Gauss Elimination Method E.X. 1 y + z = 2 2x + 3z = 5 x + y + z = 5 Solution: Augmented matrix = 0 1 1 2 2 0 3 5 1 1 1 3 ? Interchange 1 st and 2 nd equation 2 0 3 5 0 1 1 2 1 1 1 3Read More
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