110notes - Linear Algebra Dave Penneys August 7, 2008 2...

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Unformatted text preview: Linear Algebra Dave Penneys August 7, 2008 2 Contents 1 Background Material 7 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Vector Spaces 21 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Linear Combinations, Span, and (Internal) Direct Sum . . . . . . . . . . . . 24 2.3 Linear Independence and Bases . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Finitely Generated Vector Spaces and Dimension . . . . . . . . . . . . . . . 31 2.5 Existence of Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Linear Transformations 39 3.1 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Kernel and Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Polynomials 51 4.1 The Algebra of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.5 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . 59 4.6 The Polynomial Functional Calculus . . . . . . . . . . . . . . . . . . . . . . 60 5 Eigenvalues, Eigenvectors, and the Spectrum 63 5.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 The Characteristic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.4 The Minimal Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3 6 Operator Decompositions 77 6.1 Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Matrix Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.4 Nilpotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.5 Generalized Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.6 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7 Canonical Forms 89 7.1 Cyclic Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Cyclic Subspaces ....
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This note was uploaded on 10/19/2009 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at University of California, Berkeley.

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110notes - Linear Algebra Dave Penneys August 7, 2008 2...

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