Page 3 - 35Linear Algebra

Contents

1 What is Linear Algebra? 9
1.1 Organizing Information . . . . . . . . . . . . . . . . . . . . . . 9
1.2 What are Vectors? . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 What are Linear Functions? . . . . . . . . . . . . . . . . . . . 15
1.4 So, What is a Matrix? . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Matrix Multiplication is Composition of Functions . . . 25
1.4.2 The Matrix Detour . . . . . . . . . . . . . . . . . . . . 26
1.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Systems of Linear Equations 37
2.1 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Augmented Matrix Notation . . . . . . . . . . . . . . . 37
2.1.2 Equivalence and the Act of Solving . . . . . . . . . . . 40
2.1.3 Reduced Row Echelon Form . . . . . . . . . . . . . . . 40
2.1.4 Solution Sets and RREF . . . . . . . . . . . . . . . . . 45
2.2 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Elementary Row Operations . . . . . . . . . . . . . . . . . . . 52
2.3.1 EROs and Matrices . . . . . . . . . . . . . . . . . . . . 52
2.3.2 Recording EROs in (M|I ) . . . . . . . . . . . . . . . . 54
2.3.3 The Three Elementary Matrices . . . . . . . . . . . . . 56
2.3.4 LU, LDU, and PLDU Factorizations . . . . . . . . . . 58
2.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 61

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