Linear System of Equations

1. Linear System of Equations MGT 4850 Spring 2008 University of Lethbridge

2. Definition A linear equation in the variables x1, x2, . . . , xn is an equation of the form a1x1 + a2x2 + ... + anxn = b where the coefficients a1, a2, . . . , an and term b of the right-hand side are given constants.

3. Example 1.1 x + 2y = 5 4x + y = 6

4. Example 1.2 x + y + z = 4 2x + 2y + 5z = 11 4x + 6y + 8z = 24

6. Set A collection of objects, members of the set. ∅ denotes the empty set, i.e., the set with no members. a ∈ A means “a is a member of the set A.” Set Symbols A = B means “the set A is equal to the set B.” A ⊆ B means “A is a subset of B.” A ⊂ B means “A is a proper subset of B.”

7. Union and intersection • Let A = {0, 1, 3} and B = {0, 1, 2, 4}. Then A ∪ ∅ = A, A ∩ ∅ = ∅, A ∪ B = {0, 1, 2, 3, 4}, A ∩ B = {0, 1}, A − B = {3}.

8. Natural Numbers

9. The set of integers

10. Rational Numbers

11. Real Numbers • There is one more problem to overcome. How do we solve a system like x2 + 1 = 0 C of complex numbers. i2 = −1 (skip from p.12 to p.21)

12. Gaussian Elimination: 4x + 4y = 20 2x − y = 1 • Multiply the first equation by 1/4 to obtain x + y = 5 2x − y = 1

13. Gaussian Elimination • Now, multiply a copy of the first equation by −2 and add it to the second. x + y = 5 0x − 3y = −9. This part is called “forward solving.”

14. Gaussian Elimination • then work backward y = −9/−3 = 3. • Use the first equation to solve for x: x = 5− y = 5− 3 = 2.

15. Matrix Notation

16. Definition A matrix is a rectangular array of numbers. If a matrix has m rows and n columns, then the size of the matrix is said to be m×n. If the matrix is 1 × n or m × 1, it is called a vector. If m = n, then it is called a square matrix of order n. Finally, the number that occurs in the ith row and jth column is called the (i, j)th entry of the matrix.