Solving Linear Equations - Advanced Engineering Math - Lecture Slides, Slides of Engineering Mathematics

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Solving linear equations

Linear Equation in n variables

• a 1 x 1 + a 2 x 2 + … + an xn = c
  • a 1 , a 2 , …, an are called coefficients (real numbers).
  • x 1 , x 2 ,…, xn are variables (or indeterminates).
  • c is a constant term (real number).
• Example
  • 2x + 3y – 4z = 0.
• Non-example
  • x^2 + y^2 = 1

System of linear equations

• A system of linear equations (or linear system) is a collection of one or more linear equations. - for example:
• A solution is a list of numbers ( s 1 , s 2 , …, sn ) which satisfies all equalities after substituting xi by si , for i = 1 , 2 ,…, n.
• The set of all solutions is called the solution set.

Nutrition problem

• Find a combination of food A, B, C and D in order to satisfy the nutrition requirement exactly.
• Let xA , xB , xC and xD be the amount of food A, B, C and D respectively.

Food A Food B Food C Food D Requirement Protein 9 8 3 3 5 Carbohydrate 15 11 1 4 5 Vitamin A 0.02 0.003 0.01 0.006 0. Vitamin C 0.01 0.01 0.005 0.05 0.

Review of set notation

• Set of Greek letters = {,,,,,,,,,, ,,,,,,,,,,,,,}
• Set of prime numbers = {2,3,5,7,11,13,17,23,29,31,37,41, …}
• Sphere with radius r centered at origin = { ( x , y,z ) : x^2 + y^2 + z^2 = r^2 }

finite

Countably infinite

Uncountably infinite Pronounced as “such that”

Examples of solution sets

{ (x,y): ax + by = c }

-1 -0.

0 0.

1

(^0) -0.

-5 1

0

5

y x

z

x

y

Classification

Linear System

Inconsistent (no solution) Consistent

Tasks:Determine whether a linear Unique solution Infinitely manysolutions

system is consistent. If yes, find all solutions.

Short-hand notation using matrix

(2 rows, 4 columns)

Usually called the augmented matrix (4 rows, 3 columns)

Notation: row and column vectors

Row vector (^) Column vector

row 1 row 2 row 3 row 4

col. 1 (^) col. 2 col. 3 col. 4 (^) col. 5

THE ROW’S POINT OF VIEW

Illustration – row interchange

Illustration – Multiply by constant

 (^2)  2

How to solve?

• Idea: Apply the three kinds of information- lossless elementary row operations, and transform the linear system into one which is easier to solve.

Gaussian elimination

• Step 0: Write the linear system in matrix format
• Step 1: Try to transform the matrix into upper triangular form
• Step 2: Solve for the variables one by one, in backward order