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Maths Chapters

# Maths Chapters - MATH1119 Notes – By Eric Hua Contents...

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Unformatted text preview: MATH1119 Notes – By Eric Hua Contents Chapter 1. Linear Equations in Linear Algebra 2 1.1 Systems of Linear Equations . . . . . . . . . . . . . . . 2 1.2 Row reduction and echelon forms . . . . . . . . . . . 4 1.3 Vector Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Equation A~x = ~ b . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Solution sets of linear systems . . . . . . . . . . . . . . 11 1.7 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Introduction to Linear Transformation . . . . . . . 13 1.9 The matrix of a linear transformation . . . . . . . . 14 1.10 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 2. Matrix Algebra 17 2.1 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 The Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . 19 2.3 Characterization of Invertible Matrices . . . . . . 20 2.6 The Leontief Input-output Model . . . . . . . . . . . 21 2.8 Subspaces of R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.9 Dimension and Rank . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 3. Determinants 26 3.1 Introduction to Determinants . . . . . . . . . . . . . . . 26 3.2 Properties of Determinants . . . . . . . . . . . . . . . . . 27 3.3 Cramer’s rule, volume and linear transforma- tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Chapter 4. Vector Spaces 31 4.9 Applications to Markov Chains . . . . . . . . . . . . . 31 1 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....
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Maths Chapters - MATH1119 Notes – By Eric Hua Contents...

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