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Complete E-book Pdf notes of Linear Algebra with diagram and examples
Neeraj Yadav

Complete E-book Pdf notes of Linear Algebra with diagram and examples

Neeraj Yadav | 25-Dec-2015 |
Systems of Linear Equations , The Simplex Method , Vectors in Space , N-Vectors , Vector Spaces , LINEAR TRANSFORMATIONS , Matrices , Subspaces and Spanning Sets , Linear Independence , Basis and Dimension , Eigenvalues and Eigenvectors , Diagonalization , Orthonormal Bases and Complements , Diagonalizing Symmetric Matrices , Kernel , Range , Nullity , Rank , Least squares and Singular Values ,

Hi friends, here Neeraj Yadav uploaded notes for Linear Algebra with title Complete E-book Pdf notes of Linear Algebra with diagram and examples. You can download this lecture notes, ebook by clicking on the below file name or icon.

This E-book contain following topics (almost all)-

1 What is Linear Algebra? 13
1.1 What Are Vectors? . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 What Are Linear Functions? . . . . . . . . . . . . . . . . . . . 15
1.3 What is a Matrix? . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 The Matrix Detour . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Systems of Linear Equations 35
2.1 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.1 Augmented Matrix Notation . . . . . . . . . . . . . . . 35
2.1.2 Equivalence and the Act of Solving . . . . . . . . . . . 38
2.1.3 Reduced Row Echelon Form . . . . . . . . . . . . . . . 38
2.1.4 Solution Sets and RREF . . . . . . . . . . . . . . . . . 43
2.2 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Elementary Row Operations . . . . . . . . . . . . . . . . . . . 50
2.3.1 EROs and Matrices . . . . . . . . . . . . . . . . . . . . 50
2.3.2 Recording EROs in (MjI ) . . . . . . . . . . . . . . . . 52
2.3.3 The Three Elementary Matrices . . . . . . . . . . . . . 54
2.3.4 LU, LDU, and LDPU Factorizations . . . . . . . . . . 56
2.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 Solution Sets for Systems of Linear Equations . . . . . . . . . 61
2.5.1 The Geometry of Solution Sets: Hyperplanes . . . . . . 62
 
2.5.2 Particular Solution+Homogeneous Solutions . . . . . 63
2.5.3 Solutions and Linearity . . . . . . . . . . . . . . . . . . 64
2.6 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 The Simplex Method 69
3.1 Pablo's Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Graphical Solutions . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Dantzig's Algorithm . . . . . . . . . . . . . . . . . . . . . . . 73
3.4 Pablo Meets Dantzig . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Vectors in Space, n-Vectors 79
4.1 Addition and Scalar Multiplication in Rn . . . . . . . . . . . . 80
4.2 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Directions and Magnitudes . . . . . . . . . . . . . . . . . . . . 84
4.4 Vectors, Lists and Functions: RS . . . . . . . . . . . . . . . . 90
4.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Vector Spaces 97
5.1 Examples of Vector Spaces . . . . . . . . . . . . . . . . . . . . 98
5.1.1 Non-Examples . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Other Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Linear Transformations 107
6.1 The Consequence of Linearity . . . . . . . . . . . . . . . . . . 107
6.2 Linear Functions on Hyperplanes . . . . . . . . . . . . . . . . 109
6.3 Linear Dierential Operators . . . . . . . . . . . . . . . . . . . 110
6.4 Bases (Take 1) . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 114
7 Matrices 117
7.1 Linear Transformations and Matrices . . . . . . . . . . . . . . 117
7.1.1 Basis Notation . . . . . . . . . . . . . . . . . . . . . . 117
7.1.2 From Linear Operators to Matrices . . . . . . . . . . . 123
7.2 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3 Properties of Matrices . . . . . . . . . . . . . . . . . . . . . . 129
7.3.1 Associativity and Non-Commutativity . . . . . . . . . 136
 
7.3.2 Block Matrices . . . . . . . . . . . . . . . . . . . . . . 138
7.3.3 The Algebra of Square Matrices . . . . . . . . . . . . 139
7.3.4 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.5.1 Three Properties of the Inverse . . . . . . . . . . . . . 146
7.5.2 Finding Inverses (Redux) . . . . . . . . . . . . . . . . . 147
7.5.3 Linear Systems and Inverses . . . . . . . . . . . . . . . 149
7.5.4 Homogeneous Systems . . . . . . . . . . . . . . . . . . 149
7.5.5 Bit Matrices . . . . . . . . . . . . . . . . . . . . . . . . 150
7.6 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.7 LU Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.7.1 Using LU Decomposition to Solve Linear Systems . . . 155
7.7.2 Finding an LU Decomposition. . . . . . . . . . . . . . 157
7.7.3 Block LDU Decomposition . . . . . . . . . . . . . . . . 160
7.8 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 161
8 Determinants 163
8.1 The Determinant Formula . . . . . . . . . . . . . . . . . . . . 163
8.1.1 Simple Examples . . . . . . . . . . . . . . . . . . . . . 163
8.1.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . 164
8.2 Elementary Matrices and Determinants . . . . . . . . . . . . . 168
8.2.1 Row Swap . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.2.2 Row Multiplication . . . . . . . . . . . . . . . . . . . . 170
8.2.3 Row Addition . . . . . . . . . . . . . . . . . . . . . . . 171
8.2.4 Determinant of Products . . . . . . . . . . . . . . . . . 173
8.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.4 Properties of the Determinant . . . . . . . . . . . . . . . . . . 180
8.4.1 Determinant of the Inverse . . . . . . . . . . . . . . . . 183
8.4.2 Adjoint of a Matrix . . . . . . . . . . . . . . . . . . . . 183
8.4.3 Application: Volume of a Parallelepiped . . . . . . . . 185
8.5 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 186
9 Subspaces and Spanning Sets 189
9.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.2 Building Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 191
9.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 196
 
10 Linear Independence 197
10.1 Showing Linear Dependence . . . . . . . . . . . . . . . . . . . 198
10.2 Showing Linear Independence . . . . . . . . . . . . . . . . . . 201
10.3 From Dependent Independent . . . . . . . . . . . . . . . . . . 202
10.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 203
11 Basis and Dimension 207
11.1 Bases in Rn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.2 Matrix of a Linear Transformation (Redux) . . . . . . . . . . 212
11.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 215
12 Eigenvalues and Eigenvectors 219
12.1 Invariant Directions . . . . . . . . . . . . . . . . . . . . . . . . 221
12.2 The Eigenvalue{Eigenvector Equation . . . . . . . . . . . . . . 226
12.3 Eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
12.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 231
13 Diagonalization 235
13.1 Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . 235
13.2 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 236
13.3 Changing to a Basis of Eigenvectors . . . . . . . . . . . . . . . 240
13.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 242
14 Orthonormal Bases and Complements 247
14.1 Properties of the Standard Basis . . . . . . . . . . . . . . . . . 247
14.2 Orthogonal and Orthonormal Bases . . . . . . . . . . . . . . . 249
14.3 Relating Orthonormal Bases . . . . . . . . . . . . . . . . . . . 250
14.4 Gram-Schmidt & Orthogonal Complements . . . . . . . . . . 253
14.4.1 The Gram-Schmidt Procedure . . . . . . . . . . . . . . 256
14.5 QR Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 257
14.6 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . 259
14.7 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 264
15 Diagonalizing Symmetric Matrices 269
15.1 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 273
16 Kernel, Range, Nullity, Rank 277
16.1 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
16.2 Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
 
16.2.1 One-to-one and Onto . . . . . . . . . . . . . . . . . . 281
16.2.2 Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
16.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
16.4 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 290
17 Least squares and Singular Values 295
17.1 Projection Matrices . . . . . . . . . . . . . . . . . . . . . . . . 298
17.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . 300
17.3 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 304
A List of Symbols 307
B Fields 309
C Online Resources 311
D Sample First Midterm 313
E Sample Second Midterm 323
F Sample Final Exam 333
G Movie Scripts 361
G.1 What is Linear Algebra? . . . . . . . . . . . . . . . . . . . . . 361
G.2 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . 361
G.3 Vectors in Space n-Vectors . . . . . . . . . . . . . . . . . . . . 371
G.4 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
G.5 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 377
G.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
G.7 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
G.8 Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . 397
G.9 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . 398
G.10 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . 401
G.11 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 403
G.12 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 409
G.13 Orthonormal Bases and Complements . . . . . . . . . . . . . . 415
G.14 Diagonalizing Symmetric Matrices . . . . . . . . . . . . . . . . 422
G.15 Kernel, Range, Nullity, Rank . . . . . . . . . . . . . . . . . . . 424
G.16 Least Squares and Singular Values . . . . . . . . . . . . . . . . 426

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