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Linear Algebra pdf

# Linear Algebra pdf - Linear Algebra in Twenty Five Lectures...

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Unformatted text preview: Linear Algebra in Twenty Five Lectures Tom Denton and Andrew Waldron March 9, 2010 1 Contents 1 What is Linear Algebra? 6 2 Gaussian Elimination 10 2.1 Notation for Linear Systems . . . . . . . . . . . . . . . . . . . 10 2.2 Reduced Row Echelon Form . . . . . . . . . . . . . . . . . . . 12 3 Elementary Row Operations 16 4 Solution Sets for Systems of Linear Equations 21 4.1 Non-Leading Variables . . . . . . . . . . . . . . . . . . . . . . 21 5 Vectors in Space, n-Vectors 26 5.1 Directions and Magnitudes . . . . . . . . . . . . . . . . . . . . 28 6 Vector Spaces 32 7 Linear Transformations 37 8 Matrices 42 9 Properties of Matrices 47 9.1 Block Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 47 9.2 The Algebra of Square Matrices . . . . . . . . . . . . . . . . 48 10 Inverse Matrix 53 10.1 Three Properties of the Inverse . . . . . . . . . . . . . . . . . 53 10.2 Finding Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . 54 10.3 Linear Systems and Inverses . . . . . . . . . . . . . . . . . . . 55 10.4 Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . 55 10.5 Bit Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 11 LU Decomposition 59 11.1 Using LU Decomposition to Solve Linear Systems . . . . . . . 60 11.2 Finding an LU Decomposition. . . . . . . . . . . . . . . . . . 61 11.3 Block LU Decomposition . . . . . . . . . . . . . . . . . . . . . 63 2 12 Elementary Matrices and Determinants 66 12.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 12.2 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . 70 13 Elementary Matrices and Determinants II 74 14 Properties of the Determinant 80 14.1 Determinant of the Inverse . . . . . . . . . . . . . . . . . . . . 83 14.2 Adjoint of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . 83 14.3 Application: Volume of a Parallelepiped . . . . . . . . . . . . 85 15 Eigenvalues and Eigenvectors 87 15.1 Invariant Directions . . . . . . . . . . . . . . . . . . . . . . . . 87 16 Eigenvalues and Eigenvectors II 93 16.1 Eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 17 Subspaces and Spanning Sets 98 17.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 17.2 Building Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 99 18 Linear Independence 103 19 Basis and Dimension 109 19.1 Bases in R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 20 Diagonalization 115 20.1 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 119 21 Orthonormal Bases 122 21.1 Relating Orthonormal Bases . . . . . . . . . . . . . . . . . . . 125 22 Diagonalizing Symmetric Matrices 130 23 Kernel, Range, Nullity, Rank 135 24 Gram-Schmidt and Orthogonal Complements 140 24.1 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . 142 25 Least Squares 145 3 Preface These linear algebra lecture notes are designed to be presented as twenty five, fifty minute lectures suitable for sophomores likely to use the material for...
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Linear Algebra pdf - Linear Algebra in Twenty Five Lectures...

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