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Linear Algebra
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1-Systems of Linear Equations 2-Row Reduction and Echelon Forms 3- Vector Equations 4-The Matrix Equation Ax = b 5-Homogeneous and Nonhomogeneous Systems 6-Matrix Algebra ......and more.
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MATH 233 - Linear Algebra ILecture Notes
Cesar O. Aguilar
Department of Mathematics
SUNY Geneseo
,
, Lecture 0
Contents
1 Systems of Linear Equations 1
1.1 What is a system of linear equations? . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Solving linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Geometric interpretation of the solution set . . . . . . . . . . . . . . . . . . 8
2 Row Reduction and Echelon Forms 11
2.1 Row echelon form (REF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Reduced row echelon form (RREF) . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . 17
3 Vector Equations 19
3.1 Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 The linear combination problem . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 The span of a set of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 The Matrix Equation Ax = b 31
4.1 Matrix-vector multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Matrix-vector multiplication and linear combinations . . . . . . . . . . . . . 33
4.3 The matrix equation problem . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Homogeneous and Nonhomogeneous Systems 41
5.1 Homogeneous linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Nonhomogeneous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Linear Independence 49
6.1 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 The maximum size of a linearly independent set . . . . . . . . . . . . . . . . 53
7 Introduction to Linear Mappings 57
7.1 Vector mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.2 Linear mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.3 Matrix mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3
, CONTENTS
8 Onto, One-to-One, and Standard Matrix 67
8.1 Onto Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8.2 One-to-One Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.3 Standard Matrix of a Linear Mapping . . . . . . . . . . . . . . . . . . . . . . 71
9 Matrix Algebra 75
9.1 Sums of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.3 Matrix Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
10 Invertible Matrices 83
10.1 Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.2 Computing the Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 85
10.3 Invertible Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
11 Determinants 89
11.1 Determinants of 2 × 2 and 3 × 3 Matrices . . . . . . . . . . . . . . . . . . . . 89
11.2 Determinants of n × n Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 93
11.3 Triangular Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
12 Properties of the Determinant 97
12.1 ERO and Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
12.2 Determinants and Invertibility of Matrices . . . . . . . . . . . . . . . . . . . 100
12.3 Properties of the Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . 100
13 Applications of the Determinant 103
13.1 The Cofactor Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
13.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
13.3 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
14 Vector Spaces 109
14.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
14.2 Subspaces of Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
15 Linear Maps 117
15.1 Linear Maps on Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 117
15.2 Null space and Column space . . . . . . . . . . . . . . . . . . . . . . . . . . 121
16 Linear Independence, Bases, and Dimension 125
16.1 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
16.2 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
16.3 Dimension of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
17 The Rank Theorem 133
17.1 The Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4
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