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note - MATH1104 Notes By Eric Hua Contents Chapter 1 Linear...

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MATH1104 Notes – By Eric Hua Contents Chapter 1. Linear Equations in Linear Algebra 3 1.1 Systems of Linear Equations . . . . . . . . . . . . . . . 3 1.2 Row reduction and echelon forms . . . . . . . . . . . 5 1.3 Vector Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Equation A~x = ~ b . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Solution sets of linear systems . . . . . . . . . . . . . . 11 1.6 Applications of Linear Systems . . . . . . . . . . . . . 13 1.7 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . 15 1.8 Introduction to Linear Transformation . . . . . . . 15 1.9 The matrix of a linear transformation . . . . . . . . 16 1.10 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 2. Matrix Algebra 20 2.1 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 The Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . 22 2.3 Characterization of Invertible Matrices . . . . . . 23 2.8 Subspaces of R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.9 Dimension and Rank . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 3. Determinants 28 3.1 Introduction to Determinants . . . . . . . . . . . . . . . 28 3.2 Properties of Determinants . . . . . . . . . . . . . . . . . 29 3.3 Cramer’s rule, volume and linear transforma- tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Chapter 5. Eigenvalues and Eigenvectors 33 5.1 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . 33 5.2 The Characteristic Equation . . . . . . . . . . . . . . . . 34 5.3 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1
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Appendix B: Complex Numbers . . . . . . . . . . . . . . . . 37 5.5 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 39 Chapter 6. Orthogonality and Least Squares 41 6.1 Inner Product, Length and Orthogonality . . . . 41 6.2 Orthogonal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.3 Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . 46 2
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Chapter 1. Linear Equations in Linear Algebra 1.1 Systems of Linear Equations Definition 1 A linear equation in variables x 1 , x 2 , . . . , x n has the form a 1 x 1 + a 2 x 2 + a 3 x 3 + · · · + a n x n = d where the numbers a 1 , . . . , a n R are the equation’s coefficients and d R is the constant. An n -tuple ( s 1 , s 2 , . . . , s n ) R n is a solution of, or satisfies, that equation if substituting the numbers s 1 , . . . , s n for the variables gives a true statement: a 1 s 1 + a 2 s 2 + . . . + a n s n = d . A system of linear equations a 1 , 1 x 1 + a 1 , 2 x 2 + · · · + a 1 ,n x n = d 1 a 2 , 1 x 1 + a 2 , 2 x 2 + · · · + a 2 ,n x n = d 2 . . . a m, 1 x 1 + a m, 2 x 2 + · · · + a m,n x n = d m has the solution ( s 1 , s 2 , . . . , s n ) if that n -tuple is a solution of all of the equations in the system. Finding the set of all solutions is solving the system. Example 1 The ordered pair ( - 1 , 5) is a solution of this system. 3 x 1 + 2 x 2 = 7 - x 1 + x 2 = 6 In contrast, (5,-1) is not a solution. Definition 2 If we have two linear systems and they have the same solution set then the two linear systems are called equivalent. Theorem 1 The linear system has, 1. no solution 2. one solution 3. infinitely many solutions. In case 1, the linear system is called inconsistent. In case 2 or 3, the linear system is called consistent. 3
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Example 2 The system 3 x 1 + 2 x 2 = 7 - x 1 + x 2 = 6 has only one solution (-1,5). Example 3 The system x 1 + 2 x 2 = 7 - 2 x 1 - 4 x 2 = - 14 has infinite solutions (7-2k, k). Example 4 The system x 1 + 2 x 2 = 7 - 2 x 1 - hx 2 = k has no solution when h = 4 and k 6 = - 14 ; one solution when h 6 = 4 ; infinite solutions when h = 4 and k = - 14 . Matrices Definition 3 An m × n ( m by n ) matrix A with m rows and n columns with entries in R is a rectangular array of the form A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n .
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