This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MAT223H1a.doc Page 1 of 38 Lecture #1 – Tuesday, January 6, 2004 1.1 S OLUTIONS AND E LEMENTARY O PERATIONS Example Create a diet from fish and meal that contains 193g of proteins and 83g of carbohydrate. We know that fish contains 70% protein and 10% carbohydrate, and meal contains 30% protein and 60% carbohydrate. • Assume that the diet contains x g of fish and y g of meal, we obtain & ¡ ¢ = + = + 83 6 . 1 . 193 3 . 7 . y x y x . Definition A linear equation is an equation of the form b x a x a n n = + + ... 1 1 where: • x 1 ,…, x n are variables; • a 1 ,…, a n are real numbers called coefficients; • b is the constant term. Examples 1) c by ax = + . 2) 2 3 3 2 1 = + x x x . 3) 1 2 2 2 1 = + x x – not a linear equation. Definition A finite collection of linear equations in the variables x 1 ,…, x n is called a system of linear equations . Examples 1) & ¡ ¢ = = 3 1 2 1 x x . 2) & ¡ ¢ = = + 3 14 4 2 2 1 2 1 x x x x Definition Given a linear equation b x a x a n n = + + ... 1 1 , a sequence of n real numbers s 1 ,…, s n is called a solution to the linear equation if b s a s a n n = + + ... 1 1 . Similarly, this also applies to a given system of linear equations. Example Given ( ) & ¡ ¢ = = 3 1 2 1 1 x x S and ( ) & ¡ ¢ = = + 3 14 4 2 2 1 2 1 2 x x x x S , what is the solution to ( S 1 ) and ( S 2 )? • (1, 3) is a solution of ( S 1 ). • (1, 3) is also a solution of ( S 2 ) because () ( ) 14 3 4 1 2 = + and () ( ) 3 1 3 = . MAT223H1a.doc Page 2 of 38 Example • ( ) & ¡ ¢ = + = + 2 1 3 y x y x S has no solution. Example Prove that for any s , t in R, £ £ £ & £ £ £ ¡ ¢ = = = + = s s t s s s t s s 2 2 1 2 3 4 3 2 1 is a solution to the system & ¡ ¢ = + + + = + 2 1 3 4 3 2 1 4 3 2 1 x x x x x x x x . • 1 3 2 1 2 3 2 2 1 3 2 3 3 4 3 2 1 = + + + = ¤ ¥ ¦ § ¨ © + + = + t t s s s s t s t s s s s s . • 2 2 2 1 2 3 2 2 1 2 3 4 3 2 1 = + + + + = + + + + = + + + t t s s s s t s t s s s s s . Definition s , t are called parameters . s 1 ,…, s 4 described this way is said to be given in parametric form and is called the general solution of the system. Remarks • When only 2 variables are involved, solution to systems of linear equations can be described geometrically because a linear equation c by ax L = + : is a straight line if a , b are not both 0. ( ) 2 1 , s s P is in L if it is a solution of c by ax = + . • If there are two linear equations, c by ax L = + : 1 and f ey dx L = + : 2 , then the solution to the system ( ) & ¡ ¢ = + = + f ey dx c by ax S is the intersection of L 1 and L 2 . • The solution of (S) is ( ) 2 1 , s s where ( ) 2 1 2 1 , L L s s P ∩ = . MAT223H1a.doc Page 3 of 38 • The solution of ( S ) are given by the ( ) 2 1 , s s such that c bs as = + 2 1 (and this implies that f es ds = + 2 1 )....
View
Full
Document
This note was uploaded on 10/12/2011 for the course MAT 223 taught by Professor Uppa during the Fall '09 term at University of Toronto.
 Fall '09
 Uppa
 Linear Algebra, Algebra

Click to edit the document details