Math 75 Linear Algebra Class Notes Prof. Erich Holtmann For use with Elementary Linear Algebra, 7th ed., Larson Revised 21-Nov-2015 p. i Contents Chapter 1: Systems of Linear Equations 1 1.1 Introduction to Systems of Equations. 1 1.2 Gaussian Elimination and Gauss-Jordan Elimination. 9 1.3 Applications of Systems of Linear Equations. 15 Chapter 2: Matrices. 19 2.1 Operations with Matrices. 19 2.2 Properties of Matrix Operations. 23 2.3 The Inverse of a Matrix. 27 2.4 Elementary Matrices. 31 2.5 Applications of Matrix Operations. 37 Chapter 3: Determinants. 41 3.1 The Determinant of a Matrix. 41 3.2 Determinants and Elementary Operations. 45 3.3 Properties of Determinants. 51 3.4 Applications of Determinants. 55 Chapter 4: Vector Spaces. 61 8.1 Complex Numbers (Optional). 61 8.2 Conjugates and Division of Complex Numbers (Optional). 67 4.1 Vectors in Rn. 71 4.2 Vector Spaces. 77 4.3 Subspaces of Vector Spaces. 82 4.4 Spanning Sets and Linear Independence. 86 4.5 Basis and Dimension. 97 4.6 Rank of a Matrix and Systems of Linear Equations. 105 4.7 Coordinates and Change of Basis. 117 Chapter 5: Inner Product Spaces. 121 5.1 Length and Dot Product in Rn. 121 5.2 Inner Product Spaces. 129 5.3 Orthogonal Bases: Gram-Schmidt Process. 137 5.4 Mathematical Models and Least Squares Analysis (Optional). 145 5.5 Applications of Inner Product Spaces (Optional). 151 8.3 Polar Form and De Moivre's Theorem. (Optional) 157 8.4 Complex Vector Spaces and Inner Products. 163 p. i Chapter 6: Linear Transformations. 169 6.1 Introduction to Linear Transformations. 169 6.2 The Kernel and Range of a Linear Transformation. 175 6.3 Matrices for Linear Transformations. 183 6.4 Transition Matrices and Similarity. 191 6.5 Applications of Linear Transformations. 193 Chapter 7: Eigenvalues and Eigenvectors. 201 7.1 Eigenvalues and Eigenvectors. 201 7.2 Diagonalization. 209 7.3 Symmetric Matrices and Orthogonal Diagonalization. 215 7.4 Applications of Eigenvalues and Eigenvectors. 223 8.5 Unitary and Hermitian Spaces. 223 p. ii Chapter 1: Systems of Linear Equations 1.1 Introduction to Systems of Equations. Chapter 1: Systems of Linear Equations 1.1 Introduction to Systems of Equations. Objective: Recognize and solve mn systems of linear equations by hand using Gaussian elimination and back-substitution. a1x1 + a2x2 + … + anxn = b is a linear equation in standard form in n variables xi. The first nonzero coefficient ai is the leading coefficient. The constant term is b. Compare to the familiar forms of linear equation s in two variables y = mx + b and x = a. Example: Linear and Nonlinear Equations 2 (sin ) x1 – 4x2 = e sin x1 + 2x2 – 3x3 = 0 Linear Nonlinear An mn system of linear equations is a set of m linear equations in n unknowns. Example: Systems of Two Equations in Two Variables Solve and graph each 22 system. a. 2x – y = 1 b. 2x – y = 1 c. 2x – y = 1 x + y = 5 –4x + 2y = –2 –4x + 2y = 5 For a system of linear equations, exactly one of the following is true. 1) The system has exactly one solution (consistent, nonsingular system). 2) The system has infinitely many solutions (consistent, singular system). Use a free parameter or free variable (or several free parameters) to represent the solution set. 3) The system has no solution (inconsistent, singular system). p. 1 Chapter 1: Systems of Linear Equations 1.1 Introduction to Systems of Equations. To solve mn systems of linear equations (when m and n are large) we use a procedure called Gaussian elimination to find an equivalent system of equations in row-echelon form. Then we use back-substitution to solve for each variable. Row-echelon form means that the leading coefficients of 1 (called “pivots”) and the zero terms below them form a stair-step pattern. You could walk downstairs from the top left. You might have to move more than one column to the right to reach the next step, but you never have to step down more than one row at a time. 1x1 5x2 2x3 1x4 3x5 7 1x 1 Row-echelon form 3 1x4 4x5 8 1x5 9 1x1 5x2 2x3 1x4 3x5 7 1x 1 Row-echelon form 3 0 0 0 0 1x1 5x2 2x3 1x4 3x5 7 1x 3x 2x 1 Not row-echelon form 3 4 5 1x3 2x4 4x5 8 1x4 3x5 9 The goal of Gaussian elimination is to find an equivalent system that is in row-echelon form. The three operations you can use during Gaussian elimination are 1) Swap the order of two equations. 2) Multiply an equation on both sides by a non-zero constant. 3) Add a multiple of one equation to another equation. In Gaussian elimination, you start with Equation 1 (the first equation of your mn system). 1) Find the leading coefficient in the current equation. (Sometimes you need to swap equations in this step.) 2) Eliminate the coefficients of the corresponding variable in all of the equations below the current equation. 3) Move down to the next equation and go back to Step 1. Repeat until you run out of equations or you run out of variables. Solve using back-substitution: solve the last equation for the leading variable, then substitute into the preceding (i.e. second-to-last) equation and solve for its leading variable, then substitute into the preceding equation and solve for its leading variable, etc. Variables that are not leading variables are free parameters, and we often set them equal to t, s, …. p. 2 Chapter 1: Systems of Linear Equations 1.1 Introduction to Systems of Equations. Examples: Gaussian Elimination and Back-Substitution on 33 Systems of Linear Equations. p. 3 Chapter 1: Systems of Linear Equations 1.1 Introduction to Systems of Equations. p. 4 Chapter 1: Systems of Linear Equations 1.1 Introduction to Systems of Equations. p. 5 Chapter 1: Systems of Linear Equations 1.1 Introduction to Systems of Equations. p. 6 Chapter 1: Systems of Linear Equations 1.1 Introduction to Systems of Equations. Example: Chemistry Application Write and solve a system of linear equations for the chemical reaction (x1)CH4 + (x2)O2 (x3)CO2 + (x4)H2O Solution: write a separate equation for each element, showing the balance of that element. C: 1x1 + 0x2 = 1x3 + 0x4 so 1x1 + 0x2 – 1x3 + 0x4 = 0 H: 4x1 + 0x2 = 0x3 + 2x4 so 4x1 + 0x2 + 0x3 – 2x4 = 0 O: 0x1 + 2x2 = 2x3 + 1x4 so 0x1 + 2x2 – 2x3 – 1x4 = 0 p. 7 Chapter 1: Systems of Linear Equations 1.2 Gaussian Elimination and Gauss-Jordan Elimination. 1.2 Gaussian Elimination and Gauss-Jordan Elimination. Objective: Use matrices and Gauss-Jordan elimination to solve mn systems of linear equations by hand and with software. Objective: Use matrices and Gaussian elimination with back-substitution to solve mn systems of linear equations by hand and with software. Use row-echelon form or reduced row-echelon form to determine the number of solutions of a homogeneous system of linear equations, and (if applicable) the number of free parameters. A matrix is a rectangular array of numbers, called matrix a a a a entries, arranged in horizontal rows and vertical columns. 11 12 13 1n a a a a Matrices are denoted by capital letters; matrix entries are 21 22 23 2n denoted by lowercase letters with two indices. In a given A a31 a32 a33 a3n matrix entry a , the first index i is the row, and the second ij index j is the column. The entries a , a , a , … compose the 11 22 33 a a a a main diagonal. If m = n then A is called a square matrix. m1 m 2 m 3 mn a11 x1 a12 x2 a13 x3 a1n x n b1 a x a x a x a x b A linear system 21 1 22 2 23 3 2n n 2 am1 x 1 am 2 x 2 am 3 x 3 amn xn b m can represented either by a coefficient matrix A and a column vector b a11 a 12 a13 a1n b1 a a a a b 21 22 23 2n 2 A a31 a 32 a33 a3n and b b3 am1 a m 3 a m 3 amn bm or by an augmented matrix M, which I will sometimes write as [A | b] a11 a12 a13 a1n b 1 a a a a b 21 22 23 2n 2 M a31 a32 a33 a3n b 3 am1 a m 3 a m 3 amn bm (The book, Mathematica, and the calculator do not display the dotted vertical line.) To create M in Mathematica from A and b, type m=Join[a,b,2] To create M on the TI-89 from A and b, type Matrixaugment(AB.M p. 9 Chapter 1: Systems of Linear Equations 1.2 Gaussian Elimination and Gauss-Jordan Elimination. In a similar manner to that used for an mn system of linear equations, we can use a Gaussian elimination on the coefficient side A of the augmented matrix [A | b] to find an equivalent augmented matrix [U | c] in row-echelon form. Then we use back-substitution to solve for each variable. U is called an upper triangular matrix because all non-zero entries are on or above the main diagonal. row-echelon form row-echelon form not row-echelon form 1 5 2 1 3 1 5 2 1 3 1 5 2 1 3 0 0 1 0 5 0 0 1 0 5 0 0 1 3 2 0 0 0 1 4 0 0 0 0 0 0 0 1 2 4 0 0 0 0 2 0 0 0 0 0 0 0 0 1 3 Example: Use Gaussian elimination and back-substitution to solve.