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# Week 1 Notes: Sections 1.1-1.2 MATH 1554 K3

Georgia Tech

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This 6 page Class Notes was uploaded by awesomenotes on Friday January 22, 2016. The Class Notes belongs to MATH 1554 K3 at Georgia Institute of Technology taught by Stavros Garoufalidis in Spring 2016. Since its upload, it has received 49 views. For similar materials see Linear Algebra in Mathematics (M) at Georgia Institute of Technology.

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Date Created: 01/22/16

1.1 Systems of Linear Equations Linear equation – equation written in the form “a x +a x +…+a x =b” 1 1 2 2 n n where b & coefficients a ,1,a arn real or complex numbers, usually known in advance System of Linear Equations (linear system) – a collection of one or more linear equations involving the same variables Solution – a list (1 ,2 ,…,n ) of numbers that makes each equation a true statement when the values s …s a1e snbstituted for x x , 1… n respectively Solution set – set of all possible solutions -Equivalent systems have the same solution set A system of linear equations has: 1. No solution (inconsistent system) 2. Exactly one solution (consistent system) 3. Infinitely many solutions (consistent system) Matrix Notation Essential information of a linear system can be recorded compactly in a rectangular array called a matrix Augmented matrix – consists of the coefficient matrix of a system with an added column containing the constants from the right sides of the equations Size- m x n; m = # of rows, n = # of columns (rows always expressed first) Solving a Linear System Elementary Row Operations: 1. Replacement (add one row to a multiple of another row) Replace one row by the sum of itself and a multiple of another row 2. Interchange Interchange two rows 3. Scaling Multiply all entries in a row by a nonzero constant Note: row operations are reversible Row Equivalent Two matrices who have a sequence of elementary row operation that transforms one matrix into the other Two Fundamental Questions About a Linear System: 1. Is the system consistent; that is, does at least one solution exist? 2. If a solutions exists, is the solution unique? 1.2 Row Reduction and Echelon Forms Nonzero row or column- contains at least one nonzero entry Leading entry- refers to the leftmost nonzero entry (in a nonzero row) 3 Properties of Echelon Form (Row Echelon Form)ectangular Matrices: 1. All nonzero rows are above any rows of all zeros 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it 3. All leading entries in a column below a leading entry are zeros 2 Additional Properties for Reduced Echelon Form (Row Reduced Echelon Form) 4. The leading entry in each nonzero row in 1 5. Each leading 1 is the only nonzero entry in its column (0’s above and below leading 1) Echelon matrix – a matrix in echelon form Reduced echelon matrix – a matrix in reduced echelon form Uniqueness of the Reduced Echelon Form Each matrix is row equivalent to one and only one reduced echelon matrix Pivot Positions Pivot position – a location in a matrix A that corresponds to a leading 1 in the reduced echelon form of A Pivot – a nonzero # in a pivot position that is used as needed to create zeros via row operations Row Reduction Algorithm 1. Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top 2. Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position 3. Use row replacement operations to create zeros in all positions below the pivot 4. Cover (or ignore) the row containing the pivot position and cover all rows, if any, above it. Apply steps 1-3 to the submatrix that remains. Repeat the process until there are no more nonzero rows to modify To Produce A Matrix in Reduced Echelon Form: 5. Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by a scaling operation Solutions of Linear Systems Basic variable – a variable in a linear system that corresponds to a pivot column in the coefficient matrix Free variable – any variable in a linear system that is not a basic variable The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when applied to the augmented matrix of a linear system Parametric Descriptions of Solution Sets Parametric descriptions of solution sets – free variables act as parameters Solving a system – finding a parametric description of the solution set or determining the set is empty Back-Substitution The best strategy to solve a system is use only the reduced echelon form Existence and Uniqueness Theorem A linear system is consistent if and only if there rightmost column of the augmented matric is not a pivot column- that is, if and only if an echelon form of the augmented matrix has no row of the form [0…0 b] (with b ≠ 0) If a linear system is consistent, then the solution set contains either (1) a unique solution, when there are no free variables, or (2) infinitely many solutions, when there is at least one free variable Using Row Reduction to Solve a Linear System: 1. Write the augmented matric of the system 2. Use the row reduction algorithm to obtain an equivalent augmented matric in echelon form. Decide whether the system is consistent. If there is no solution, stop 3. Continue row reduction to obtain reduced echelon form 4. Write the system of equations corresponding to the matrix obtained in step 3 5. Rewrite each nonzero equation from step 4 so that its one basic variable is expressed in terms of any free variables appearing in the equations 1.1 Systems of Linear Equations Linear equation – equation written in the form “a x +a x +…+a x =b” where 1 1 2 2 n n b & coefficients a 1…,a nre real or complex numbers, usually known in advance System of Linear Equations (linear system) – a collection of one or more linear equations involving the same variables Solution – a list (1 2s ,…,n ) of numbers that makes each equation a true statement when the values s …s 1re nubstituted for x x , r1…pentively Solution set – set of all possible solutions -Equivalent systems have the same solution set A system of linear equations has: 1. No solution (inconsistent system) 2. Exactly one solution (consistent system) 3. Infinitely many solutions (consistent system) Matrix Notation Essential information of a linear system can be recorded compactly in a rectangular array called a matrix Augmented matrix – consists of the coefficient matrix of a system with an added column containing the constants from the right sides of the equations Size- m x n; m = # of rows, n = # of columns (rows always expressed first) Solving a Linear System Elementary Row Operations: 1. Replacement (add one row to a multiple of another row) Replace one row by the sum of itself and a multiple of another row 2. Interchange Interchange two rows 3. Scaling Multiply all entries in a row by a nonzero constant Note: row operations are reversible Row Equivalent Two matrices who have a sequence of elementary row operation that transforms one matrix into the other Two Fundamental Questions About a Linear System: 1. Is the system consistent; that is, does at least one solution exist? 2. If a solution exists, is the solution unique? 1.2 Row Reduction and Echelon Forms 1 Nonzero row or column- contains at least one nonzero entry Leading entry- refers to the leftmost nonzero entry (in a nonzero row) 3 Properties of Echelon Form (Row Echelon Form)ctangular Matrices: 1. All nonzero rows are above any rows of all zeros 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it 3. All leading entries in a column below a leading entry are zeros 2 Additional Properties for Reduced Echelon Form (Row Reduced Echelon Form) 4. The leading entry in each nonzero row in 1 5. Each leading 1 is the only nonzero entry in its column (0’s above and below leading 1) Echelon matrix – a matrix in echelon form Reduced echelon matrix – a matrix in reduced echelon form Theorem 1 Uniqueness of the Reduced Echelon Form Each matrix is row equivalent to one and only one reduced echelon matrix. Pivot Positions Pivot position – a location in a matrix A that corresponds to a leading 1 in the reduced echelon form of A Pivot – a nonzero # in a pivot position that is used as needed to create zeros via row operations Row Reduction Algorithm 1. Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top 2. Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position 3. Use row replacement operations to create zeros in all positions below the pivot 4. Cover (or ignore) the row containing the pivot position and cover all rows, if any, above it. Apply steps 1-3 to the submatrix that remains. Repeat the process until there are no more nonzero rows to modify To Produce A Matrix in Reduced Echelon Form: 5. Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by a scaling operation Solutions of Linear Systems Basic variable – a variable in a linear system that corresponds to a pivot column in the coefficient matrix Free variable – any variable in a linear system that is not a basic variable 2 The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when applied to the augmented matrix of a linear system Parametric Descriptions of Solution Sets Parametric descriptions of solution sets – free variables act as parameters Solving a system – finding a parametric description of the solution set or determining the set is empty Back-Substitution The best strategy to solve a system is use only the reduced echelon form Theorem 2 Existence and Uniqueness Theorem A linear system is consistent if and only if there rightmost column of the augmented matric is not a pivot column- that is, if and only if an echelon form of the augmented matrix has no row of the form 0 …0 b [ withbnonzero If a linear system is consistent, then the solution set contains either (1) a unique solution, when there are no free variables, or (2) infinitely many solutions, when there is at least one free variable. Using Row Reduction to Solve a Linear System: 1. Write the augmented matric of the system 2. Use the row reduction algorithm to obtain an equivalent augmented matric in echelon form. Decide whether the system is consistent. If there is no solution, stop 3. Continue row reduction to obtain reduced echelon form 4. Write the system of equations corresponding to the matrix obtained in step 3 5. Rewrite each nonzero equation from step 4 so that its one basic variable is expressed in terms of any free variables appearing in the equations *The symbol ~ between matrices denotes row equivalencies 3

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