110notes - Linear Algebra Dave Penneys 2 Contents 1...

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Unformatted text preview: Linear Algebra Dave Penneys June 23, 2008 2 Contents 1 Background Material 7 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Vector Spaces 17 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Linear Combinations, Span, and Linear Independence . . . . . . . . . . . . . 19 2.3 Finitely Generated Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Existence of Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 (Internal) Direct Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Linear Transformations 33 3.1 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Kernel and Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5 Idempotents and Matrix Decomposition . . . . . . . . . . . . . . . . . . . . 42 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Polynomials 47 4.1 The Algebra of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.4 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . 52 4.5 The Polynomial Functional Calculus . . . . . . . . . . . . . . . . . . . . . . 53 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 5 Eigenvalues, Eigenvectors, and Diagonalization 55 5.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3 The Minimal Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.4 The Primary Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . 59 5.5 Trace and Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.6 The Characteristic Polynomial I . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6 Canonical Forms 67 6.1 Cyclic Subspaces and Rational Canonical Form . . . . . . . . . . . . . . . . 67 6.2 The Characteristic Polynomial II . . . . . . . . . . . . . . . . . . . . . . . .The Characteristic Polynomial II ....
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This note was uploaded on 07/19/2008 for the course MATH 110 taught by Professor Gurevitch during the Summer '08 term at Berkeley.

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110notes - Linear Algebra Dave Penneys 2 Contents 1...

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