Lect2 - Chapter 1. Matrices and Systems of Equations...

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Unformatted text preview: Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x1 + 2x2 + x3 = Elementary Row Operations - Example 3 Example. Solve 3x1 − x2 − 3x3 = −1 2x1 + 3x2 + x3 = 4 1 3 2 2 −1 3 1 −3 1 3 −1 4 Solution. The augmented matrix is Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x1 + 2x2 + x3 = Elementary Row Operations - Example 3 Example. Solve 3x1 − x2 − 3x3 = −1 2x1 + 3x2 + x3 = 4 1 3 2 2 −1 3 1 −3 1 3 −1 4 Solution. The augmented matrix is Apply the operations: −3R1 + R2 , Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x1 + 2x2 + x3 = Elementary Row Operations - Example 3 Example. Solve 3x1 − x2 − 3x3 = −1 2x1 + 3x2 + x3 = 4 1 3 2 2 −1 3 1 −3 1 3 −1 4 Solution. The augmented matrix is Apply the operations: −3R1 + R2 , −2R1 + R3 , Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x1 + 2x2 + x3 = Elementary Row Operations - Example 3 Example. Solve 3x1 − x2 − 3x3 = −1 2x1 + 3x2 + x3 = 4 1 3 2 2 −1 3 1 −3 1 3 −1 4 Solution. The augmented matrix is Apply the operations: −3R1 + R2 , −2R1 + R3 , − 1 R2 + R3 , 7 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x1 + 2x2 + x3 = Elementary Row Operations - Example 3 Example. Solve 3x1 − x2 − 3x3 = −1 2x1 + 3x2 + x3 = 4 1 3 2 2 −1 3 1 −3 1 3 −1 4 Solution. The augmented matrix is Apply the operations: −3R1 + R2 , −2R1 + R3 , − 1 R2 + R3 , 7 1 0 0 2 1 −6 −1 7 3 the augmented matrix becomes −7 0 −10 4 −7 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x1 + 2x2 + x3 = Elementary Row Operations - Example 3 Example. Solve 3x1 − x2 − 3x3 = −1 2x1 + 3x2 + x3 = 4 1 3 2 2 −1 3 1 −3 1 3 −1 4 Solution. The augmented matrix is Apply the operations: −3R1 + R2 , −2R1 + R3 , − 1 R2 + R3 , 7 1 0 0 2 1 −6 −1 7 3 the augmented matrix becomes −7 0 −10 4 −7 We can read the solution from it Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations x1 + 2x2 + x3 = Elementary Row Operations - Example 3 Example. Solve 3x1 − x2 − 3x3 = −1 2x1 + 3x2 + x3 = 4 1 3 2 2 −1 3 1 −3 1 3 −1 4 Solution. The augmented matrix is Apply the operations: −3R1 + R2 , −2R1 + R3 , − 1 R2 + R3 , 7 1 0 0 2 1 −6 −1 7 3 the augmented matrix becomes −7 0 −10 4 −7 We can read the solution from it - write it back into equations. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry Pick a row with a nonzero entry to be pivotal row Perform row operations to eliminate other entries in the column Repeat the steps to the remaining smaller matrix. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry Pick a row with a nonzero entry to be pivotal row Perform row operations to eliminate other entries in the column Repeat the steps to the remaining smaller matrix. 1 3 2 2 −1 3 1 −3 1 3 −1 4 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry Pick a row with a nonzero entry to be pivotal row Perform row operations to eliminate other entries in the column Repeat the steps to the remaining smaller matrix. 1 3 2 2 −1 3 1 −3 1 3 −1 4 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry Pick a row with a nonzero entry to be pivotal row Perform row operations to eliminate other entries in the column Repeat the steps to the remaining smaller matrix. 1 3 2 2 −1 3 1 −3 1 3 1 −1 −→ 0 4 0 2 −7 −1 1 −6 −1 3 −10 −2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry Pick a row with a nonzero entry to be pivotal row Perform row operations to eliminate other entries in the column Repeat the steps to the remaining smaller matrix. 1 3 2 2 −1 3 1 −3 1 3 1 −1 −→ 0 4 0 2 −7 −1 1 −6 −1 3 −10 −2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Methodology Locate the 1st column containing a nonzero entry Pick a row with a nonzero entry to be pivotal row Perform row operations to eliminate other entries in the column Repeat the steps to the remaining smaller matrix. 1 3 2 2 −1 3 1 −3 1 3 1 1 2 −1 −→ 0 −7 −6 4 0 −1 − 1 1 1 2 0 − 7 −6 −→ 1 0 0 −7 3 −10 −2 3 −10 4 −7 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 0 1 2 3 −1 1 4 1 −1 1 1 −2 1 1 −2 2 0 6 −1 3 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 0 1 2 3 −1 1 4 1 −1 1 1 −2 1 1 −2 2 0 6 −1 3 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 2 3 1 −1 4 1 1 −1 1 −2 1 1 −2 2 6 0 −1 3 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 0 0 1 −1 2 −2 1 −1 −1 −5 1 1 −4 −1 6 0 −13 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 0 0 1 −1 2 −2 1 −1 −1 −5 1 1 −4 −1 6 0 −13 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 0 0 1 −1 2 −2 1 −1 −1 −5 1 1 4 −1 6 0 −13 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 0 0 1 −1 0 0 1 −1 −3 −3 1 1 −2 −3 6 0 −13 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 0 0 1 −1 0 0 1 −1 −3 −3 1 1 −2 −3 6 0 −13 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 0 0 1 −1 0 0 1 −1 −3 −3 1 1 −2 −3 6 0 −13 −15 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 0 0 1 −1 0 0 1 −1 −3 0 1 1 −2 −1 6 0 −13 −2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Elementary Row Operations - Example Example. Solve −x2 − x3 + x4 = = = = 0 6 −1 3 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 Solution. The augmented matrix is 1 0 0 0 1 −1 0 0 1 −1 −3 0 1 1 −2 −1 6 0 −13 −2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Homework 2 Reading Leon (7th edition): p.8 - p.11 Leon (8th edition): p.7 - p.10 Homework 2 Leon (7th edition): Chapter 1 - Section 1 Qn. 8, 6(f)-(h). Leon (8th edition): Section 1.1 Qn. 8, 6(f)-(h). Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 1 Augmented matrix in Triangular form - tell solution immediately. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 1 Augmented matrix in Triangular form - tell solution immediately. Example. Solve the system −x2 − x3 + x4 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = = = = 0 6 −1 3 if its augmented matrix is reduced to 1 0 0 0 1 1 0 0 1 1 1 0 1 −1 2 3 1 6 0 13 3 2 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 1 Augmented matrix in Triangular form - tell solution immediately. Example. Solve the system −x2 − x3 + x4 x1 + x2 + x3 + x4 2x1 + 4x2 + x3 − 2x4 3x1 + x2 − 2x3 + 2x4 = = = = 0 6 −1 3 if its augmented matrix is reduced to 1 0 0 0 1 1 0 0 1 1 1 0 1 −1 2 3 1 6 0 13 3 2 The solution set is {(−4, −1, 3, 2)}. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 2 Example. Solve the system x1 + x2 + x3 + x4 + x5 −x1 − x2 + x5 −2x1 − 2x2 + 3x5 x3 + x4 + 3x5 x1 + x2 + 2x3 + 2x4 + 4x5 = = = = = 1 −1 1 3 4 if its augmented matrix is reduced to 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 2 1 0 0 1 0 3 0 0 Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 2 Example. Solve the system x1 + x2 + x3 + x4 + x5 −x1 − x2 + x5 −2x1 − 2x2 + 3x5 x3 + x4 + 3x5 x1 + x2 + 2x3 + 2x4 + 4x5 = = = = = 1 −1 1 3 4 if its augmented matrix is reduced to 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 2 1 0 0 1 0 3 0 0 The solution set is {(4 − t , t , −6 − s, s, 3) : t , s ∈ }. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Lead & Free variables There are two types of variables in the 2nd example. Lead variables & Free variables A variable corresponding to a leading one is called a lead variable All remaining variables are called free variables Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Lead & Free variables There are two types of variables in the 2nd example. Lead variables & Free variables A variable corresponding to a leading one is called a lead variable All remaining variables are called free variables Remark A system has infinitely many solutions if and only if it has free variable(s). Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 3 Example. The augmented matrix of the system x1 + x2 + x3 + x4 + x5 −x1 − x2 + x5 −2x1 − 2x2 + 3x5 x3 + x4 + 3x5 x1 + x2 + 2x3 + 2x4 + 4x5 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 2 1 0 0 = = = = = 1 −1 1 −1 1 is reduced to 1 0 3 −4 −3 Solve the system. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form - Example 3 Example. The augmented matrix of the system x1 + x2 + x3 + x4 + x5 −x1 − x2 + x5 −2x1 − 2x2 + 3x5 x3 + x4 + 3x5 x1 + x2 + 2x3 + 2x4 + 4x5 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 2 1 0 0 = = = = = 1 −1 1 −1 1 is reduced to 1 0 3 −4 −3 Solve the system. Consistent system A linear system is consistent if it has solution(s); otherwise, it is inconsistent. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form Row Echelon Form (i) The first nonzero entry in each nonzero row is 1. (ii) The number of leading zeros of nonzero rows increases from the top to the bottom. (iii) All zero rows, if any, are at the bottom. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form Row Echelon Form (i) The first nonzero entry in each nonzero row is 1. (ii) The number of leading zeros of nonzero rows increases from the top to the bottom. (iii) All zero rows, if any, are at the bottom. 1 0 0 4 1 0 2 1 3 , 0 1 0 2 0 0 3 1 1 , 0 0 0 3 0 0 1 1 0 0 3 0 Examples: Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Row Echelon Form Row Echelon Form (i) The first nonzero entry in each nonzero row is 1. (ii) The number of leading zeros of nonzero rows increases from the top to the bottom. (iii) All zero rows, if any, are at the bottom. 1 0 0 2 1 2 3 1 3 1 0 1 3 , 0 0 1 , 0 0 1 3 01 000 0000 2 4 6 1 0 1 0 0 0 0 0 1 3 , , 0 0 1 3 010 001 0010 4 Examples: Non-examples: Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Homework 3 Reading Leon (7th edition): p.13 - p.15 Leon (8th edition): p.11 - p.13 Homework 3 Leon (7th edition): Chapter 1 Section 2 Qn. 1, 2(c)-(f), 3(c)-(f), 4. Leon (8th edition): Section 1.2 Qn. 1, 2(c)-(f), 3(c)-(f), 4. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Gaussian Elimination Gaussian Elimination - a process : augmented matrix row operation − − − − − −−→ − − − − −− row echelon form Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Gaussian Elimination Gaussian Elimination - a process : augmented matrix row operation − − − − − −−→ − − − − −− row echelon form Row echelon form Solutions of a system Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Gaussian Elimination Gaussian Elimination - a process : augmented matrix row operation − − − − − −−→ − − − − −− row echelon form Row echelon form Solutions of a system x+y x−y −x + 2y = = = 1 by Gaussian Elimination. 3 −4 Example. Solve the system Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Gaussian Elimination Gaussian Elimination - a process : augmented matrix row operation − − − − − −−→ − − − − −− row echelon form Row echelon form Solutions of a system x+y x−y −x + 2y 1 0 = = = 1 by Gaussian Elimination. 3 −4 1 −1 . 0 Example. Solve the system Ans. A row echelon form is 0 1 0 1 The solution is x = 2, y = −1. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Gaussian Elimination Gaussian Elimination - a process : augmented matrix row operation − − − − − −−→ − − − − −− row echelon form Row echelon form Solutions of a system x+y+z 2x + 3y = = 1 5 Example. Solve the system x + 2y − z = 3 by Gaussian Elimination. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Gaussian Elimination Gaussian Elimination - a process : augmented matrix row operation − − − − − −−→ − − − − −− row echelon form Row echelon form Solutions of a system x+y+z 2x + 3y 1 1 0 = = 1 5 Example. Solve the system x + 2y − z = 3 by Gaussian Elimination. 1 −2 0 1 2 1 . 1 0 0 Ans. A row echelon form is No solution. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Gaussian Elimination Gaussian Elimination - a process : augmented matrix row operation − − − − − −−→ − − − − −− row echelon form Row echelon form Solutions of a system x+y+z 2x + 3y = = 1 4 Example. Solve the system x + 2y − z = 3 by Gaussian Elimination. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Gaussian Elimination Gaussian Elimination - a process : augmented matrix row operation − − − − − −−→ − − − − −− row echelon form Row echelon form Solutions of a system x+y+z 2x + 3y 1 1 0 = = 1 4 Example. Solve the system x + 2y − z = 3 by Gaussian Elimination. 1 −2 0 1 2 0 . 1 0 0 Ans. A row echelon form is The solution is x = −1 − 3α, y = 2 + 2α, z = α, where α ∈ . Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Overdetermined & Underdetermined Systems Some Terminologies A linear system is said to be overdetermined if the number of equations is more than the number of unknowns. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Overdetermined & Underdetermined Systems Some Terminologies A linear system is said to be overdetermined if the number of equations is more than the number of unknowns. A linear system is said to be underdetermined if the number of equations is less than the number of unknowns. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Overdetermined & Underdetermined Systems Some Terminologies A linear system is said to be overdetermined if the number of equations is more than the number of unknowns. A linear system is said to be underdetermined if the number of equations is less than the number of unknowns. Questions Must an overdetermined system be inconsistent? Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Overdetermined & Underdetermined Systems Some Terminologies A linear system is said to be overdetermined if the number of equations is more than the number of unknowns. A linear system is said to be underdetermined if the number of equations is less than the number of unknowns. Questions Must an overdetermined system be inconsistent? Must an underdetermined system have infinitely many solutions? Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Homogeneous Systems Terminology A linear system is said to be homogeneous if the constants on the righthand side are all zero. Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Homogeneous Systems Terminology A linear system is said to be homogeneous if the constants on the righthand side are all zero. Questions Must a homogeneous system be consistent? Chapter 1. Matrices and Systems of Equations Math1111 Systems of Linear Equations Type of Solutions Observations (0 0 ··· 0 1) means an inconsistent system Number of lead variables = number of variables (i.e. no free variable) Each variable is uniquely determined. ∴ Exactly one solution. Free variable ⇒ infinitely many solutions Remarks. 1. The number of lead variables = the number of leading one’s. (Recall what is a lead variable.) 2. In Section 1.2.4 of Nicholson, the number of leading one’s is called the "rank of a matrix". We do not use this name in this chapter because we shall define "rank" in another way. These two definitions are equivalent (i.e. defining the same object). ...
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