Chapter 2 Simultaneous Linear Equations

1. Chapter 2Simultaneous Linear Equations

2. 2.1 Linear systems • A system of m linear equations in n variables is a set of m equations, each of which is linear in the same n variables: • A solution is a set of scalars x1 , x2 , …, xn that when substituted in the system satisfies the given equations. • A linear system can possess exactly one solution, an infinite number of solutions, or no solution. • A linear system is called consistent if it has at least one solution and inconsistent if it has no solution. • A linear system can be written in matrix form: Ax=b(see details on the board) • A linear system is called homogenous if b=0

3. 2.1 Linear systems (HW example) Modeling a real-life situation as a linear model • A manufacturer produces desks and bookcases. • Desks d require 5 hours of cutting time and 10 hours of assembling time. • Bookcases b require 15 minutes of cutting time and one hour of assembling time. • Each day, the manufacturer has available 200 hours for cutting and 500 hours for assembling. • The manufacturer wants to know how many desks and bookcases should be scheduled for completion each day to utilize all available workpower. • Show that this problem is equivalent to solving two equations in the two unknowns d and b .

4. Augmented matrix of a linear system • The matrix derived from the coefficients and constant terms of a system of linear equations is called the augmented matrix of the system. • The matrix containing only the coefficients of the system is called the coefficient matrix of the system. • System Augmented MatrixCoefficient Matrix const. y z x

5. Elementary row operations (E1) Interchange any two rows. (E2) Multiply any row by a nonzero scalar. (E3) Add a multiple of a row to another row. Two matrices are said to be row-equivalent if one can be obtained from the other by a finite sequence of elementary row operations. Row-equivalent systems have the same set of solutions.

6. Gaussian Elimination • Write the augmented matrix of the system. • Use elementary row operations to transform it to an equivalent row-reduced form. (this is most often accomplished by using (E3) with each diagonal element to create zeros in all columns directly below it, beginning with the first column) • The system associated with row-reduced matrix can be solved easily by back-substitution.

7. Gaussian Elimination: Example 1 Linear System Associated Augmented matrix R2+R1R2 (2) R3+R2R3 0.5R3R3 0.5

8. (1) (2) (3) Gaussian Elimination: Example 2 A system with no solution • Solve the system 0 = 2 … ???The original system of linear equationsis inconsistent.