Algebra linear ii

FUNDAMENTALS OF LINEAR ALGEBRA
James B. Carrell [e-mail] (July, 2005)

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Contents
1 Introduction 2 Linear Equations and Matrices 2.1 Linear equations: the beginning of algebra . . . . 2.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Matrix Addition and Vectors . . . . . . . 2.2.2 Some Examples . . . . . . . . . . . . . . . 2.2.3 Matrix Product . . . . . . . . . . . . . . . 2.3 Reduced Row Echelon Form and Row Operations 2.4 Solving Linear Systems via Gaussian Reduction . 2.4.1 Row Operations and Equivalent Systems . 2.4.2 The Homogeneous Case . . . . . . . . . . 2.4.3 The Non-homogeneous Case . . . . . . . . 2.4.4 Criteria for Consistency and Uniqueness . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . 11 15 15 18 18 19 21 23 26 26 27 29 31 36 37 37 39 40 43 46 49 50 51 53 59 59 61

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3 More Matrix Theory 3.1 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . 3.1.1 The Transpose of a Matrix . . . . . . . . . . . . . 3.1.2 The Algebraic Laws . . . . . . . . . . . . . . . . . 3.2 Elementary Matrices and Row Operations . . . . . . . . . 3.2.1 Application to Linear Systems . . . . . . . . . . . 3.3 Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 A Necessary and Sufficient Condition for Existence 3.3.2 Methods for Finding Inverses . . . . . . . . . . . . 3.3.3 Matrix Groups . . . . . . . . . . . . . . . . . . . . 3.4 The LP DU Factorization . . . . . . . . . . . . . . . . . . 3.4.1 The Basic Ingredients: L, P, D and U . . . . . . . 3.4.2 The Main Result . . . . . . . . . . . . . . . . . . . 3

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4 3.4.3 Further Uniqueness in LP DU . . . . . . 3.4.4 Further Uniqueness in LP DU . . . . . . 3.4.5 The symmetric LDU decomposition . . 3.4.6 LP DU and Reduced Row