MTH 306 Spring Term 2007
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1 MTH 306 Spring Term 2007 Lesson 3 John Lee Oregon State University (Oregon State University) 1 / 27
2 Lesson 3 Goals: Be able to solve 2 2 and 3 3 linear systems by systematic elimination of unknowns without the aid of a calculator or computer (Oregon State University) 2 / 27
3 Lesson 3 Goals: Be able to solve 2 2 and 3 3 linear systems by systematic elimination of unknowns without the aid of a calculator or computer Be able to evaluate 2 2 and 3 3 determinants without the aid of a calculator or computer (Oregon State University) 2 / 27
4 Lesson 3 Goals: Be able to solve 2 2 and 3 3 linear systems by systematic elimination of unknowns without the aid of a calculator or computer Be able to evaluate 2 2 and 3 3 determinants without the aid of a calculator or computer Understand that systematic elimination of unknowns can be applied to n n linear systems and is the basis for computer solution of large systems (Oregon State University) 2 / 27
5 Lesson 3 Goals: Be able to solve 2 2 and 3 3 linear systems by systematic elimination of unknowns without the aid of a calculator or computer Be able to evaluate 2 2 and 3 3 determinants without the aid of a calculator or computer Understand that systematic elimination of unknowns can be applied to n n linear systems and is the basis for computer solution of large systems Learn Theorems 1, 2, and 3. Be able to illustrate them geometrically for 2 2 and 3 3 linear systems (Oregon State University) 2 / 27
6 Solving 2 x 2 Linear Systems Example Solve the system x y = 3 2x + 3y = 4. Solution 1. Graph the two equations: (Oregon State University) 3 / 27
7 4 2 x y = x + 3y = 4 4 Conclusion from graph? (Oregon State University) 4 / 27
8 4 2 x y = x + 3y = 4 4 Conclusion from graph? x 1 and y 2 (Oregon State University) 4 / 27
9 Solving 2 x 2 Linear Systems the geometric view ax + by = e graphs as line L1 (S) cx + dy = f graphs as line L 2 Case I. Lines L 1 and L 2 intersect in a unique point. y x Conclusion: The system (S) has a unique solution. (Oregon State University) 5 / 27
10 Case II. Lines L 1 and L 2 are distinct parallel lines. y x Conclusion: The system (S) has a no solutions. (Oregon State University) 6 / 27
11 Case III. Lines L 1 and L 2 coincide. y x Conclusion: The system (S) has in nitely many solutions. (Oregon State University) 7 / 27
12 Fundamental Fact 1 Theorem Any system of linear algebraic equations has either a unique solution, no solution, or in nitely many solutions. The theorem holds no matter how many equations there are and no matter how many unknowns there are. (Oregon State University) 8 / 27
13 Fundamental Fact 1 Theorem Any system of linear algebraic equations has either a unique solution, no solution, or in nitely many solutions. The theorem holds no matter how many equations there are and no matter how many unknowns there are. We have established the theorem for 2 linear equations in 2 unknows. (Oregon State University) 8 / 27
14 Fundamental Fact 1 Theorem Any system of linear algebraic equations has either a unique solution, no solution, or in nitely many solutions. The theorem holds no matter how many equations there are and no matter how many unknowns there are. We have established the theorem for 2 linear equations in 2 unknows. What is the geometric proof for systems of linear equations of the form ax + by + cz = d? (Oregon State University) 8 / 27
15 Solving 2 x 2 Linear Systems the algebraic view The solutions to ax + by = e cx + dy = f where a, b, c, d, e, and f are given numbers can easily be found by elimination of unknowns. Example Use elimination of unknowns to solve x y = 3 2x + 3y = 4. Solution 2: (Oregon State University) 9 / 27
16 Example Use elimination of unknowns to solve u v = 3 2u + 3v = 4. Solution : (Oregon State University) 10 / 27
17 Example Use elimination of unknowns to solve u v = 3 2u + 3v = 4. Solution : This is the same system as in the previous example but with the names of the unknowns changed. So u = 1 and v = 2. The point? (Oregon State University) 10 / 27
18 Example Use elimination of unknowns to solve u v = 3 2u + 3v = 4. Solution : This is the same system as in the previous example but with the names of the unknowns changed. So u = 1 and v = 2. The point? The solution to a linear system does not depend on the symbols used to represent the unknows. So those symbols aren t needed in the solution of the system. (Oregon State University) 10 / 27
19 Example Use elimination of unknowns to solve u v = 3 2u + 3v = 4. Solution : This is the same system as in the previous example but with the names of the unknowns changed. So u = 1 and v = 2. The point? The solution to a linear system does not depend on the symbols used to represent the unknows. So those symbols aren t needed in the solution of the system. This leads to an e cient systematic way to solve linear systems. Stay tuned! (Oregon State University) 10 / 27
20 Coe cient Matrix A matrix is a rectangular array of items. For us the items will be numbers. (Oregon State University) 11 / 27
21 Coe cient Matrix A matrix is a rectangular array of items. For us the items will be numbers. The linear system has coe cient matrix ax + by = e cx + dy = f a A = c b d. (Oregon State University) 11 / 27
22 Coe cient Matrix A matrix is a rectangular array of items. For us the items will be numbers. The linear system has coe cient matrix ax + by = e cx + dy = f a A = c Convention: Capital letters are usually used to denote matrices and square brackets enclose the entries. b d. (Oregon State University) 11 / 27
23 Augmented Matrix The linear system has augmented matrix ax + by = e cx + dy = f a b e c d f. (Oregon State University) 12 / 27
24 When will a System have a Unique Solution? We will only answer this question for an n n (AKA square) system. (Oregon State University) 13 / 27
25 When will a System have a Unique Solution? We will only answer this question for an n n (AKA square) system. The 2 2 case will guide us: (S) ax + by = e graphs as line L1 cx + dy = f graphs as line L 2 (Oregon State University) 13 / 27
26 When will a System have a Unique Solution? We will only answer this question for an n n (AKA square) system. The 2 2 case will guide us: (S) ax + by = e graphs as line L1 cx + dy = f graphs as line L 2 What property of these lines guarantees that we are in Case 1 Lines L 1 and L 2 intersect in a unique point? (Oregon State University) 13 / 27
27 When will a System have a Unique Solution? We will only answer this question for an n n (AKA square) system. The 2 2 case will guide us: (S) ax + by = e graphs as line L1 cx + dy = f graphs as line L 2 What property of these lines guarantees that we are in Case 1 Lines L 1 and L 2 intersect in a unique point? Answer: (Oregon State University) 13 / 27
28 When will a System have a Unique Solution? We will only answer this question for an n n (AKA square) system. The 2 2 case will guide us: (S) ax + by = e graphs as line L1 cx + dy = f graphs as line L 2 What property of these lines guarantees that we are in Case 1 Lines L 1 and L 2 intersect in a unique point? Answer: (Oregon State University) 13 / 27
29 When will a System have a Unique Solution? We will only answer this question for an n n (AKA square) system. The 2 2 case will guide us: (S) ax + by = e graphs as line L1 cx + dy = f graphs as line L 2 What property of these lines guarantees that we are in Case 1 Lines L 1 and L 2 intersect in a unique point? Answer:ad bc 6= 0. (Oregon State University) 13 / 27
30 Determinants The number ad bc is called the determinant of the system ax + by = e cx + dy = f or of its coe cient matrix A. Notation for determinants: a b c d = ad bc. Important: (Oregon State University) 14 / 27
31 Determinants The number ad bc is called the determinant of the system ax + by = e cx + dy = f or of its coe cient matrix A. Notation for determinants: a b c d = ad bc. Important: A matrix is a rectangular array of items. (Oregon State University) 14 / 27
32 Determinants The number ad bc is called the determinant of the system ax + by = e cx + dy = f or of its coe cient matrix A. Notation for determinants: a b c d = ad bc. Important: A matrix is a rectangular array of items. A determinant is a (single) number. (Oregon State University) 14 / 27
33 Determinants The number ad bc is called the determinant of the system ax + by = e cx + dy = f or of its coe cient matrix A. Notation for determinants: a b c d = ad bc. Important: A matrix is a rectangular array of items. A determinant is a (single) number. We also express the determinant of A by jaj or by det (A). (Oregon State University) 14 / 27
34 Fundamental Fact 2 Theorem An n n linear system of algebraic equations has a unique solution if and only if its determinant is not zero. You can learn when an m n linear system of algebraic equations has a unique solution if you take a course in linear algebra. (Oregon State University) 15 / 27
35 Fundamental Fact 2 Theorem An n n linear system of algebraic equations has a unique solution if and only if its determinant is not zero. You can learn when an m n linear system of algebraic equations has a unique solution if you take a course in linear algebra. Of course we don t yet know what the determinant is for an n n linear system when n > 2. What is it? (Oregon State University) 15 / 27
36 3 x 3 Determinants The determinant of the system of three linear equations 8 < ax + by + cz = j dx + ey + fz = k : gx + hy + iz = l in the three unknowns x, y, z and where a, b, c,..., j, k, and l are given numbers is a b c d e f = aei ahf bdi + bgf + cdh cge. g h i How do you remember that! Read on. (Oregon State University) 16 / 27
37 Laplace Expansions a b c d e f g h i = a e h f i b d g f i + c d g e h is called the (Laplace) expansion of the determinant by its rst row. Do you see the pattern? (Oregon State University) 17 / 27
38 Laplace Expansions a b c d e f g h i = a e h f i b d g f i + c d g e h is called the (Laplace) expansion of the determinant by its rst row. Do you see the pattern? Cross out the row and column containing a on the left to get the 2 2 determinant e f h i which multiplies a on the right. (Oregon State University) 17 / 27
39 Laplace Expansions a b c d e f g h i = a e h f i b d g f i + c d g e h is called the (Laplace) expansion of the determinant by its rst row. Do you see the pattern? Cross out the row and column containing a on the left to get the 2 2 determinant e f h i which multiplies a on the right. The second and third terms on the right are formed in the corresponding way. (Oregon State University) 17 / 27
40 Laplace Expansions a b c d e f g h i = a e h f i b d g f i + c d g e h is called the (Laplace) expansion of the determinant by its rst row. Do you see the pattern? Cross out the row and column containing a on the left to get the 2 2 determinant e f h i which multiplies a on the right. The second and third terms on the right are formed in the corresponding way. These terms are combined with the alternating sign pattern +,, +. (Oregon State University) 17 / 27
41 Homogeneous Linear Systems Homogeneous linear systems whose right members are all zeros when the unknowns are all moved to the left are very important. (Oregon State University) 18 / 27
42 Homogeneous Linear Systems Homogeneous linear systems whose right members are all zeros when the unknowns are all moved to the left are very important. Once again the 2 2 case will guide us: (H) ax + by = 0 cx + dy = 0. The lines L 1 and L 2 that represent the two equations in this homogeneous system both pass through the origin. Why? (Oregon State University) 18 / 27
43 Homogeneous Linear Systems Homogeneous linear systems whose right members are all zeros when the unknowns are all moved to the left are very important. Once again the 2 2 case will guide us: (H) ax + by = 0 cx + dy = 0. The lines L 1 and L 2 that represent the two equations in this homogeneous system both pass through the origin. Why? So x = 0 and y = 0 is a solution to the system. It is called the trivial solution. (Oregon State University) 18 / 27
44 Homogeneous Linear Systems Homogeneous linear systems whose right members are all zeros when the unknowns are all moved to the left are very important. Once again the 2 2 case will guide us: (H) ax + by = 0 cx + dy = 0. The lines L 1 and L 2 that represent the two equations in this homogeneous system both pass through the origin. Why? So x = 0 and y = 0 is a solution to the system. It is called the trivial solution. Graph L 1 and L 2 to see there are two possibilities: (Oregon State University) 18 / 27
45 Either L 1 and L 2 are distinct lines: y x Only the trivial solution Which happens when? or (Oregon State University) 19 / 27
46 Either L 1 and L 2 are distinct lines: y x Only the trivial solution or Which happens when? Answer: determinant of the system is 6= 0 (Oregon State University) 19 / 27
47 L 1 and L 2 coincide: y x Nontrivial solutions Which happens when? (Oregon State University) 20 / 27
48 L 1 and L 2 coincide: y x Nontrivial solutions Which happens when? Answer: determinant of the system is = 0 (Oregon State University) 20 / 27
49 L 1 and L 2 coincide: y x Nontrivial solutions Which happens when? Answer: determinant of the system is = 0 These observations are summarized next: (Oregon State University) 20 / 27
50 Fundamental Fact 3 Theorem An n n homogeneous linear system of algebraic equations has nontrivial solutions if and only if its determinant is zero. Corollary An n n homogeneous linear system of algebraic equations has only the trivial solution if and only if its determinant is not zero. You can learn the corresponding results for an m n homogeneous linear system if you take a course in linear algebra. (Oregon State University) 21 / 27
51 Examples of Systematic Elimination All of the fundamental facts covered in Lesson 3 can be established by systematic elimination of unknowns. This process is usually called Gauss(ian) elimination. (Oregon State University) 22 / 27
52 Examples of Systematic Elimination All of the fundamental facts covered in Lesson 3 can be established by systematic elimination of unknowns. This process is usually called Gauss(ian) elimination. We will use Gauss elimination to reduce a linear system to an equivalent system (a system that has the same set of solutions) whose augmented matrix is in row echelon form. Such systems are easy to solve by back substitution. (Oregon State University) 22 / 27
53 Examples of Systematic Elimination All of the fundamental facts covered in Lesson 3 can be established by systematic elimination of unknowns. This process is usually called Gauss(ian) elimination. We will use Gauss elimination to reduce a linear system to an equivalent system (a system that has the same set of solutions) whose augmented matrix is in row echelon form. Such systems are easy to solve by back substitution. Later we will nd the inverse of a square matrix. Then we will use Gauss elimination to even more systematically and put the related system of equations that enable us to nd the inverse of the matrix in reduced row echelon form. (Oregon State University) 22 / 27
54 Example Solve x + y = 1 4x 3y = 17. (Oregon State University) 23 / 27
55 Example Solve x + y = 1 4x 3y = 17 Think rst! and x + y = 4 4x 3y = 9. (Oregon State University) 24 / 27
56 Example Solve 8 < : 3x 11y + 9z = 2 x 4y + 4z = 1 2x + y 5z = 3. (Oregon State University) 25 / 27
57 Example Solve 8 < : 5x + y 3z = 5 3x + 2y 4z = 4 6x + 3y 3z = 2. (Oregon State University) 26 / 27
58 Example Solve 8 < : x + y 3z = 5 2x + 3y + 5z = 0 3x 2y 8z = 5. (Oregon State University) 27 / 27
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