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Unformatted text preview: Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Objectives Consider the following simple examples: (a) x1 + x2 = 2
x1 − x2 = 2 (b) x1 + x2 = 2
x1 + x2 = 1 (c) x1 + x2 = 2 − x1 − x2 = − 2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Objectives Consider the following simple examples: (a) x1 + x2 = 2
x1 − x2 = 2 (b) x1 + x2 = 2
x1 + x2 = 1 (c) x1 + x2 = 2 − x1 − x2 = − 2 Problems: What are the possible types of solutions? Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Objectives Consider the following simple examples: (a) x1 + x2 = 2
x1 − x2 = 2 (b) x1 + x2 = 2
x1 + x2 = 1 (c) x1 + x2 = 2 − x1 − x2 = − 2 Problems: What are the possible types of solutions? How to determine the solution type? Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Objectives Consider the following simple examples: (a) x1 + x2 = 2
x1 − x2 = 2 (b) x1 + x2 = 2
x1 + x2 = 1 (c) x1 + x2 = 2 − x1 − x2 = − 2 Problems: What are the possible types of solutions? How to determine the solution type? How to see if two systems represent the same solutions? Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Objectives Consider the following simple examples: (a) x1 + x2 = 2
x1 − x2 = 2 (b) x1 + x2 = 2
x1 + x2 = 1 (c) x1 + x2 = 2 − x1 − x2 = − 2 Problems: What are the possible types of solutions? How to determine the solution type? How to see if two systems represent the same solutions? How to solve a “big” system? Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations References Leon (7th edition) Chapter 1. Sections 12 (8th edition) Chapter 1. Sections 1.11.2 DeFranza Chapter 1. Sections 1.11.2 Nicholson (1st edition) Chapter 1. Sections 1.21.3.1 (2nd edition) Chapter 1. Sections 1.21.3.1 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Terminologies Linear equation in n unknowns/variables:
a1 1 x1 + a1 2 x2 + · · · + a1 n xn = b Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Terminologies Linear equation in n unknowns/variables:
a1 1 x1 + a1 2 x2 + · · · + a1 n xn = b System of m linear equations in n unknowns: a1 1 x1 + a1 2 x2 + · · · + a1 n xn a2 1 x1 + a2 2 x2 + · · · + a2 n xn . . . = = b1 b2 . . . . . . am 1 x1 + am 2 x2 + · · · + am n xn = bm Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Terminologies Linear equation in n unknowns/variables:
a1 1 x1 + a1 2 x2 + · · · + a1 n xn = b System of m linear equations in n unknowns: a1 1 x1 + a1 2 x2 + · · · + a1 n xn a2 1 x1 + a2 2 x2 + · · · + a2 n xn . . . = = b1 b2 . . . . . . am 1 x1 + am 2 x2 + · · · + am n xn = bm Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Terminologies Linear equation in n unknowns/variables:
a1 1 x1 + a1 2 x2 + · · · + a1 n xn = b System of m linear equations in n unknowns: a1 1 x1 + a1 2 x2 + · · · + a1 n xn a2 1 x1 + a2 2 x2 + · · · + a2 n xn . . . = = b1 b2 . . . . . . am 1 x1 + am 2 x2 + · · · + am n xn = bm Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Terminologies Linear equation in n unknowns/variables:
a1 1 x1 + a1 2 x2 + · · · + a1 n xn = b System of m linear equations in n unknowns: a1 1 x1 + a1 2 x2 + · · · + a1 n xn a2 1 x1 + a2 2 x2 + · · · + a2 n xn . . . = = b1 b2 . . . . . . am 1 x1 + am 2 x2 + · · · + am n xn = bm Call it m × n (linear) system. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations More Terminologies Solution set = the collection of solution(s) of a system Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations More Terminologies Solution set = the collection of solution(s) of a system We use the bracket { · · · } to collect the solutions. Reconsider the example: Example. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations More Terminologies Solution set = the collection of solution(s) of a system We use the bracket { · · · } to collect the solutions. Reconsider the example: Example. (a) x1 + x2 = 2
x1 − x2 = 2 The solution set is {(2, 0)}. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations More Terminologies Solution set = the collection of solution(s) of a system We use the bracket { · · · } to collect the solutions. Reconsider the example: Example. (b) x1 + x2 = 2
x1 + x2 = 1 The solution set is { }, denoted by . Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations More Terminologies Solution set = the collection of solution(s) of a system We use the bracket { · · · } to collect the solutions. Reconsider the example: Example. (c)
x1 + x2 = 2 −x1 − x2 = −2 The solution set is {(a, 2 − a) : a is a real number}. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations More Terminologies Solution set = the collection of solution(s) of a system We use the bracket { · · · } to collect the solutions. Reconsider the example: Example. (c)
x1 + x2 = 2 −x1 − x2 = −2 The solution set is {(a, 2 − a) : a is a real number}. We can describe a solution set in diﬀerent ways: Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations More Terminologies Solution set = the collection of solution(s) of a system We use the bracket { · · · } to collect the solutions. Reconsider the example: Example. (c)
x1 + x2 = 2 −x1 − x2 = −2 The solution set is {(a, 2 − a) : a is a real number}. We can describe a solution set in diﬀerent ways:
{(2 − α, α) : α is a real number}, Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations More Terminologies Solution set = the collection of solution(s) of a system We use the bracket { · · · } to collect the solutions. Reconsider the example: Example. (c)
x1 + x2 = 2 −x1 − x2 = −2 The solution set is {(a, 2 − a) : a is a real number}. We can describe a solution set in diﬀerent ways:
{(2 − α, α) : α is a real number}, {(1 − t , 1 + t ) : t is a real number}. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. The equation x + 2y = 2 represents a straight line. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. The equation x + 2y = 2 represents a straight line. Solution set of x + 2y = 2 is {(2 − 2α, α) : α ∈ } Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. The equation x + 2y = 2 represents a straight line. Solution set of x + 2y = 2 is {(2 − 2α, α) : α ∈ } Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. The equation x + 2y = 2 represents a straight line. Solution set of x + 2y = 2 is {(2 − 2α, α) : α ∈ } Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. The equation x + y + z = 2 represents a plane. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. The equation x + y + z = 2 represents a plane. Solution set is {(2 − α − β , α, β ) : α, β ∈ } Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. The equation x + y + z = 2 represents a plane. Solution set is {(2 − α − β , α, β ) : α, β ∈ } Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. What does the solution of mean? x−y = 1 x + 2y = 2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Geometric Interpretation Example. What does the solution of mean? x−y = 1 x + 2y = 2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x+y+z = 2 Geometric Interpretation Example. What is geometrically meant? x−y−z = 2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x+y+z = 2 Geometric Interpretation Example. What is geometrically meant? x−y−z = 2 Solution set = {(2, −α, α) : α ∈ }. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x+y+z = 2 Geometric Interpretation Example. What is geometrically meant? x−y−z = 2 Solution set = {(2, −α, α) : α ∈ }. More variable cases can be interpreted similarly in higher dimensional space! Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Equivalent Systems Equivalent Systems Two systems involving the same variables are equivalent if they have the same solution sets. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Equivalent Systems Equivalent Systems Two systems involving the same variables are equivalent if they have the same solution sets. Example. Show that the following systems are equivalent. (a)
x1 + x2 = 2 − x1 − x2 = − 2 (b) x1 + x2 = 2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Equivalent Systems Equivalent Systems Two systems involving the same variables are equivalent if they have the same solution sets. Example. Show that the following systems are equivalent. (a)
x1 + x2 = 2 − x1 − x2 = − 2 (b) x1 + x2 = 2 Example. Show that the following systems are equivalent. (i) 3x1 + 2x2 − x3 = −2
x2 =3 2x3 = 4 (ii) 3x1 + 2x2 − x3 = −2
−3x1 − x2 + x3 = 5 3x1 + 2x2 + x3 = 2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Solving Method A system is characterized by its solution set. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Solving Method A system is characterized by its solution set. How to solve a system? Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Solving Method A system is characterized by its solution set. How to solve a system? Example. Solve 2x − y −x + 7y = 9 = −10 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Solving Method A system is characterized by its solution set. How to solve a system? Example. Solve Solution. 2x − y −x + 7y = 2x − y −x + 7y = 9 = −10 9 = −10 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Solving Method A system is characterized by its solution set. How to solve a system? Example. Solve Solution. 2x − y −x + 7y = 2x − y −x + 7y = 9 = −10 9 = −10 − − − − − −−→ − − − − −− −x + 7y 2x − y = −10 = 9 Interchange the equations Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Solving Method A system is characterized by its solution set. How to solve a system? Example. Solve Solution. 2x − y −x + 7y = 2x − y −x + 7y = 9 = −10 9 = −10 = −20 = 9 − − − − − −−→ − − − − −− −x + 7y 2x − y = −10 = 9 −2x + 14y 2x − y Multiply 2 to 1st equation Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Solving Method A system is characterized by its solution set. How to solve a system? Example. Solve Solution. 2x − y −x + 7y = 2x − y −x + 7y = 9 = −10 9 = −10 = −20 = 9 − − − − − −−→ − − − − −− −x + 7y 2x − y = −10 = 9 = −20 = −11 −2x + 14y 2x − y − − − − − −−→ − − − − −− −2x + 14y 13y Add 1st equation to the 2nd Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Solving Method A system is characterized by its solution set. How to solve a system? Example. Solve Solution. 2x − y −x + 7y = 2x − y −x + 7y = 9 = −10 9 = −10 = −20 = 9 − − − − − −−→ − − − − −− −x + 7y 2x − y = −10 = 9 = −20 = −11 −2x + 14y 2x − y − − − − − −−→ − − − − −− −2x + 14y 13y Solve for y , x by backward substitution Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix Observation The variables and equality signs do not take part. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix Observation The variables and equality signs do not take part. For a m × n system a1 1 x 1 + a1 2 x 2 + · · · + a1 n x n a2 1 x1 + a2 2 x2 + · · · + a2 n xn = = b1 b2 . . . . . .
= . . .
bm am 1 x1 + am 2 x2 + · · · + am n xn the augmented matrix is a rectangular array Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix Observation The variables and equality signs do not take part. For a m × n system a1 1 x 1 + a1 2 x 2 + · · · + a1 n x n a2 1 x1 + a2 2 x2 + · · · + a2 n xn = = b1 b2 . . . . . .
= . . .
bm am 1 x1 + am 2 x2 + · · · + am n xn a1 1 a2 1 . . . am 1 a1 2 ··· a1 n the augmented matrix is a rectangular array a2 2 . . . ··· . . . a2 n . . . am 2 ··· am n Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix Observation The variables and equality signs do not take part. For a m × n system a1 1 x 1 + a1 2 x 2 + · · · + a1 n x n a2 1 x1 + a2 2 x2 + · · · + a2 n xn = = b1 b2 . . . . . .
= . . .
bm am 1 x1 + am 2 x2 + · · · + am n xn a1 1 a2 1 . . . am 1 a1 2 ··· a1 n the augmented matrix is a rectangular array a2 2 . . . ··· . . . a2 n . . . am 2 ··· am n b1 b2 . . . bm Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix Observation The variables and equality signs do not take part. For a m × n system a1 1 x 1 + a1 2 x 2 + · · · + a1 n x n a2 1 x1 + a2 2 x2 + · · · + a2 n xn = = b1 b2 . . . . . .
= . . .
bm am 1 x1 + am 2 x2 + · · · + am n xn a1 1 a2 1 . . . am 1 a1 2 ··· a1 n the augmented matrix is a rectangular array a2 2 . . . ··· . . . a2 n . . . am 2 ··· am n b1 b2 . . . bm The vertical line  separate coeﬃcients & constant terms Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix  example Example. Write down the augmented matrix of x1 + 2x2 + x3 =3 3x1 − x2 − 3x3 2x1 + 3x2 + x3 = −1 =4 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Augmented matrix  example Example. Write down the augmented matrix of x1 + 2x2 + x3 =3 3x1 − x2 − 3x3 2x1 + 3x2 + x3 = −1 =4 Solution. The augmented matrix is 1 1 2 3 −1 −3 23 1 3 −1 4 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Homework 1 Reading Leon (7th or 8th edition): p.1  p.7 Homework 1 Leon (7th edition): Chapter 1  Section 1 Qn 1, 3, 2, 4, 5. Leon (8th edition): Section 1.1 Qn 1, 3, 2, 4, 5. Homeworks mean exercises for your drilling at home; hence, not for handin. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations 2x − y = Elementary Row Operations  Motivation
9 Example. Solve −x + 7y = −10 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations 2x − y = Elementary Row Operations  Motivation
9 Example. Solve −x + 7y = −10 2x − y Solution. − x + 7y −2x + 14y 2x − y = = −20 9 = = 9 −10 − − − − − −−→ − − − − −− − x + 7y 2x − y = = = = −20 −11 −10 9 − − − − − −−→ − − − − −− −2x + 14y 13y Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations 2x − y = Elementary Row Operations  Motivation
9 Example. Solve −x + 7y = −10 2x − y Solution. − x + 7y −2x + 14y 2x − y = = −20 9 = = 9 −10 − − − − − −−→ − − − − −− − x + 7y 2x − y = = = = −20 −11 −10 9 − − − − − −−→ − − − − −− −2x + 14y 13y In terms of augmented matrix, 2 −1 −1 7 9 −10 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations 2x − y = Elementary Row Operations  Motivation
9 Example. Solve −x + 7y = −10 2x − y Solution. − x + 7y −2x + 14y 2x − y = = −20 9 = = 9 −10 Interchange the equations − x + 7y 2x − y = = = = −20 −11 −10 9 − − − − − −−→ − − − − −− − − − − − −−→ − − − − −− −2x + 14y 13y In terms of augmented matrix, 2 −1 −1 7 −1 −→ 2 −10 9 7 −1 −10 9 Interchange rows Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations 2x − y = Elementary Row Operations  Motivation
9 Example. Solve −x + 7y = −10 2x − y Solution. − x + 7y −2x + 14y 2x − y = = −20 9 = = 9 −10 Multiply 2 to 1st equation − x + 7y 2x − y = = = = −20 −11 −10 9 − − − − − −−→ − − − − −− − − − − − −−→ − − − − −− −2x + 14y 13y In terms of augmented matrix, 2 −1 −2 2 −1 7 14 −1 −1 −→ 2 −10 −20 9 9 7 −1 Multiply 2 to 1st row −10 −→ 9 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations 2x − y = Elementary Row Operations  Motivation
9 Example. Solve −x + 7y = −10 2x − y Solution. − x + 7y −2x + 14y 2x − y = = −20 9 = = 9 −10 Add 1st equation to the 2nd − x + 7y 2x − y = = = = −20 −11 −10 9 − − − − − −−→ − − − − −− − − − − − −−→ − − − − −− −2x + 14y 13y In terms of augmented matrix, 2 −1 −2 2 −1 7 14 −1 −1 −→ 2 −10 −20 −2 −→ 9 0 9 7 −1 14 13 Add 1st row to the 2nd −10 −→ 9 −20 −11 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations The above operations are suﬃcient to solve any system of linear equations! Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations The above operations are suﬃcient to solve any system of linear equations! Elementary Row Operations
I. Interchange two rows Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations The above operations are suﬃcient to solve any system of linear equations! Elementary Row Operations
I. Interchange two rows Notation
I. Ri ↔ Rj (interchange ith & jth rows) Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations The above operations are suﬃcient to solve any system of linear equations! Elementary Row Operations
I. Interchange two rows II. Multiply a row by a nonzero real number Notation
I. Ri ↔ Rj (interchange ith & jth rows) Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations The above operations are suﬃcient to solve any system of linear equations! Elementary Row Operations
I. Interchange two rows II. Multiply a row by a nonzero real number Notation
I. Ri ↔ Rj II. αRi (interchange ith & jth rows) (Multiply ith row by α) Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations The above operations are suﬃcient to solve any system of linear equations! Elementary Row Operations
I. Interchange two rows II. Multiply a row by a nonzero real number III. Replace a row by its sum with a multiple of another row Notation
I. Ri ↔ Rj II. αRi (interchange ith & jth rows) (Multiply ith row by α) Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations The above operations are suﬃcient to solve any system of linear equations! Elementary Row Operations
I. Interchange two rows II. Multiply a row by a nonzero real number III. Replace a row by its sum with a multiple of another row Notation
I. Ri ↔ Rj II. αRi III. αRi + Rj (interchange ith & jth rows) (Multiply ith row by α) (Replace jth row by αRi + Rj ) or (Add αRi to Rj ) ...
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