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University of Toronto St. George

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MAT223H1

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MAT223H1a.doc 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. 0.7x +0.3y =193 Assume that the diet contains xg of fish and yg of meal, we obtan . 0.1x + 0.6y = 83 Definition A linear equationis an equation of the form a x +...+ a x = b where: 1 1 n n x1,, xnare variables; a ,, a are real numbers called coefficients; 1 n b is the constant term. Examples 1) ax +by = c . 2) 3x1+ x 2x =3 0 . 2 3) 2x1 + x2=1 not a linear equation. Definition A finite collection of linear equations in the vari1bles n ,, x is called a system of linear equations. Examples 1) x1=1 . x = 3 2 2x + 4x =14 2) 1 2 3 x1 x2 = 0 Definition Given a linear equation 1 1 +...+a n n = b, a sequence of n real numbers1s ,,ns is called a solution to the linear equation if a s +...+ a s = b . Similarly, this also applies to a given system of linear equations. 1 1 n n Example x1 =1 2x1 + 4x2 =14 Given (S1 and (S 2 , what is the solution to1 and (S2)? x2 = 3 3x1 x2 = 0 (1, 3) is a solution of1(S (1, 3) is also a solution of (Secause 2(1)+ 43 )=14 and 3 (1)(3)= 0. 2 Page 1 of 38 www.notesolution.com MAT223H1a.doc Example x+ y =1 ( 3 has no solution. x+ y = 2 Example s = 3 + t 1 2 s2 = s x13 x2 + x3 x4 =1 Prove that for any s, t in R is a solution to the syste . s3 =t x1+ x2 + x3+ x4 = 2 1 s4 = 2s 2 3 1 3 1 s1 s 2 s 3 s =4 + s t 3s +t 2 = +s 3s + s t =t 1 . 2 2 2 2 s + s + s + s = 3+ s t + s +t + 2s = 3 +1 + s + s s t +t =2. 1 2 3 4 2 2 2 2 Definition s, t are called parameters1 s ,4 s 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 L :ax +by = c is a straight line if a, b are not both 0. P s 1s 2) is in L if it is a solution of ax +by = c . If there are two linear equations,1L :ax +by = c and L2:dx +ey = f , then the solution to the system ax +by = c ( dx +ey = f is the intersection o1and L 2 (s1, 2) The solution of (S) is where P s1,s 2)= L1 L 2 . Page 2 of 38 www.notesolution.com MAT223H1a.doc (s , s) The solution of (S) are given by the 2 L such that as +bs = c (and this implies that 1 1 2 ds1+es 2 f ). (S) has no solution. Definition The elementary operationsre: 1) Interchange 2 equations. 2) Multiply one equation by a non-zero number. 3) Add a multiple of one equation to a different equation. Definition Two systems of linear equations are said to be equivalent iff the solutions of the systems are the same. Theorem Suppose that an elementary operation is performed on a system of linear equations, then the resulting system is equivalent to the original one. Example x+ 2y =1 Solve: . 3x y = 4 x+ 2y= 1 R x+ 2y =1 R x+ 2y =1 R 7x = 9 R =R 1 1 1 1 2 3x y 4 R2 6x 2y =8 R2 =2 R2 7x =9 R2= R1+ R2 x+ 2y =1 R2=R 1 9 9 x =7 x =7 2y = 9 = 2 y = 1 7 7 7 The solution of the system is ,1 . 7 7 Page 3 of 38 www.notesolution.com

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