MATHS 233

MATH 233 - Linear Algebra I
Lecture 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 =b31
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
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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×nMatrices ......................... 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
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Lecture 0
18 Coordinate Systems 137
18.1 Coordinates .................................... 137
18.2 Coordinate Mappings ............................... 141
18.3 Matrix Representation of a Linear Map ..................... 142
19 Change of Basis 147
19.1 Review of Coordinate Mappings on Rn..................... 147
19.2 Change of Basis .................................. 149
20 Inner Products and Orthogonality 153
20.1 Inner Product on Rn............................... 153
20.2 Orthogonality ................................... 156
20.3 Coordinates in an Orthonormal Basis ...................... 158
21 Eigenvalues and Eigenvectors 163
21.1 Eigenvectors and Eigenvalues .......................... 163
21.2 When λ= 0 is an eigenvalue ........................... 168
22 The Characteristic Polynomial 169
22.1 The Characteristic Polynomial of a Matrix ................... 169
22.2 Eigenvalues and Similarity Transformations .................. 176
23 Diagonalization 179
23.1 Eigenvalues of Triangular Matrices ....................... 179
23.2 Diagonalization .................................. 180
23.3 Conditions for Diagonalization .......................... 182
24 Diagonalization of Symmetric Matrices 187
24.1 Symmetric Matrices ................................ 187
24.2 Eigenvectors of Symmetric Matrices ....................... 188
24.3 Symmetric Matrices are Diagonalizable ..................... 188
25 The PageRank Algortihm 191
25.1 Search Engine Retrieval Process ......................... 191
25.2 A Description of the PageRank Algorithm ................... 192
25.3 Computation of the PageRank Vector ...................... 195
26 Discrete Dynamical Systems 197
26.1 Discrete Dynamical Systems ........................... 197
26.2 Population Model ................................. 197
26.3 Stability of Discrete Dynamical Systems .................... 199
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