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18.701

Algebra I

3-0-9 units · Undergraduate

18.701 - 18.702 is more extensive and theoretical than the 18.700 - 18.703 sequence. Experience with proofs necessary. 18.701 focuses on group theory, geometry, and linear algebra.

Fall 2010

Topics

  • The general linear group GL n , groups, generators
  • The symmetric group S n , subgroups
  • Subgroups of Z+, cyclic groups
  • Homomorphisms and isomorphisms
  • Cosets
  • The correspondence theorem
  • Product groups, quotient groups
  • Fields
  • Vector spaces, bases, dimension
  • Computation with bases
  • The dimension formula
  • Linear operators, eigenvectors
  • The characteristic polynomial
  • Jordan form
  • Rotations
  • Isometries
  • Finite groups of isometries
  • Discrete groups of isometries
  • Discrete groups of isometries (cont.)
  • Group operations
  • Finite rotation groups
  • The class equation
  • The icosahedral group
  • The symmetric and alternating groups
  • Symmetric and hermitian forms
  • Orthogonality
  • The projection formula
  • The spectral theorem
  • Quadrics
  • The special unitary group SU 2
  • The rotation group SO 3
  • One-parameter groups
  • One-parameter groups (cont.)
  • The lie algebra
  • Simple groups

Readings

  • Artin, M. Algebra (2nd Edition) . Addison Wesley, 2010. ISBN: 9780132413770.
    Listed reading
  • SES #: 1 · TOPICS: The general linear group GL n , groups, generators · READINGS: Chapter 1, sections 1-4; chapter 2, sections 1-2 · EXERCISES: Chapter 1, 1.7, 3.4, 4.6; chapter 2, 1.1, and 2.3
    Reading assignment
  • SES #: 2 · TOPICS: The symmetric group S n , subgroups · READINGS: Chapter 1, section 5; chapter 2, section 2 · EXERCISES: Chapter 1, 5.1, 5.2; chapter 2, 2.4, and 2.5
    Reading assignment
  • SES #: 3 · TOPICS: Subgroups of Z+, cyclic groups · READINGS: Chapter 2, sections 3-4 · EXERCISES: Chapter 2, 3.1, 3.2, 4.4, 4.5, and 4.9
    Reading assignment
  • SES #: 4 · TOPICS: Homomorphisms and isomorphisms · READINGS: Chapter 2, sections 5-7 · EXERCISES: Chapter 2, 5.3, 6.2, 6.6, 7.1, and 7.5
    Reading assignment
  • SES #: 5 · TOPICS: Cosets · READINGS: Chapter 2, sections 8-9 · EXERCISES: Chapter 2, 8.3, 8.4, 8.5, and 8.10
    Reading assignment
  • SES #: 6 · TOPICS: The correspondence theorem · READINGS: Chapter 2, section 10 · EXERCISES: Chapter 2, 9.4, 9.5, 10.1, and 10.3
    Reading assignment
  • SES #: 7 · TOPICS: Product groups, quotient groups · READINGS: Chapter 2, sections 11-12 · EXERCISES: Chapter 2, 11.3, 11.4, 12.1, and 12.5
    Reading assignment
  • SES #: 8 · TOPICS: Fields · READINGS: Chapter 3, section 1 · EXERCISES: Chapter 3, 1.2, 1.3, 1.10, and 1.11
    Reading assignment
  • SES #: 9 · TOPICS: Vector spaces, bases, dimension · READINGS: Chapter 3, sections 2-3 · EXERCISES: Chapter 3, 2.2, 3.1, 3.7, 3.8, and 4.5
    Reading assignment
  • SES #: 10 · TOPICS: Computation with bases · READINGS: Chapter 3, sections 4-5 · EXERCISES: Chapter 3, 4.2, 4.3, 4.4, 5.1, and 5.2
    Reading assignment
  • SES #: 11 · TOPICS: The dimension formula · READINGS: Chapter 4, section 2 · EXERCISES: Chapter 4, 1.1, 1.3, 1.4, 2.1, and 2.3
    Reading assignment
  • SES #: 12 · TOPICS: Linear operators, eigenvectors · READINGS: Chapter 4, sections 3-4 · EXERCISES: Chapter 4, 3.3, 4.2, 4.4, 4.6, and 4.8
    Reading assignment
  • SES #: 13 · TOPICS: The characteristic polynomial · READINGS: Chapter 4, sections 5-6 · EXERCISES: Chapter 4, 5.3, 5.5, 5.10, 6.1, and 6.4
    Reading assignment
  • SES #: 14 · TOPICS: Jordan form · READINGS: Chapter 4, section 7 · EXERCISES: Chapter 4, 6.10, 7.1, 7.3, 7.6, and 7.7
    Reading assignment
  • SES #: 15 · TOPICS: Rotations · READINGS: Chapter 5, sections 1-2 · EXERCISES: Chapter 5, 1.1, 1.2, 1.5, 2.1, and 2.2
    Reading assignment
  • SES #: 16 · TOPICS: Isometries · READINGS: Chapter 6, sections 1-3 · EXERCISES: Chapter 6, 1.1, 3.1, 3.2, 3.4, and 3.6
    Reading assignment
  • SES #: 17 · TOPICS: Finite groups of isometries · READINGS: Chapter 6, section 4 · EXERCISES: Chapter 6, 4.1, 4.2, and 4.3
    Reading assignment
  • SES #: 18 · TOPICS: Discrete groups of isometries · READINGS: Chapter 6, section 5 · EXERCISES: Chapter 6, 5.1, 5.2, 5.3, 5.5, and 5.6
    Reading assignment
  • SES #: 19 · TOPICS: Discrete groups of isometries (cont.) · READINGS: Chapter 6, section 6 · EXERCISES: Chapter 6, 5.11, 6.1, 6.12, and 6.3
    Reading assignment
  • SES #: 20 · TOPICS: Group operations · READINGS: Chapter 6, sections 7-9 · EXERCISES: Chapter 6, 7.2, 7.4, 8.3, 9.1, and 9.6
    Reading assignment
  • SES #: 21 · TOPICS: Finite rotation groups · READINGS: Chapter 6, section 12 · EXERCISES: Chapter 6, 10.1, 12.1, 12.3, 12.5, and 12.7
    Reading assignment
  • SES #: 22 · TOPICS: The class equation · READINGS: Chapter 7, sections 1-3 · EXERCISES: Chapter 7, 2.1, 2.5, 2.7, 2.17, and 3.1
    Reading assignment
  • SES #: 23 · TOPICS: The icosahedral group · READINGS: Chapter 7, section 4 · EXERCISES: 4.1, 4.2, 4.3, and 4.4
    Reading assignment
  • SES #: 24 · TOPICS: The symmetric and alternating groups · READINGS: Chapter 6, section 11; chapter 7, sections 4-5 · EXERCISES: Chapter 6, 11.9; chapter 7, 4.7, 5.1c, 5.2, and 5.3
    Reading assignment
  • SES #: 25 · TOPICS: Symmetric and hermitian forms · READINGS: Chapter 8, sections 1-3 · EXERCISES: Chapter 8, 1.1, 2.1, 3.2, 3.3, and 3.4
    Reading assignment
  • SES #: 26 · TOPICS: Orthogonality · READINGS: Chapter 8, section 4 · EXERCISES: Chapter 8, 4.3, 4.4, 4.5, 4.7, 4.9, and 4.14
    Reading assignment
  • SES #: 27 · TOPICS: The projection formula · READINGS: Chapter 8, sections 4-5 · EXERCISES: Chapter 8, 4.2, 4.15, 4.21, and 5.4
    Reading assignment
  • SES #: 28 · TOPICS: The spectral theorem · READINGS: Chapter 8, section 6 · EXERCISES: Chapter 8, 6.3, 6.6, 6.9, 6.14, and 6.18
    Reading assignment
  • SES #: 29 · TOPICS: Quadrics · READINGS: Chapter 8, section 7 · EXERCISES: Chapter 8, 6.21, 7.1, 7.2, and 7.3
    Reading assignment
  • SES #: 30 · TOPICS: The special unitary group SU 2 · READINGS: Chapter 9, sections 1-3 · EXERCISES: Chapter 9, 1.2, 1.5, 2.1, 3.1, and 3.4
    Reading assignment
  • SES #: 31 · TOPICS: The rotation group SO 3 · READINGS: Chapter 9, section 4 · EXERCISES: Chapter 9, 4.1, 4.2, 4.4a, and 4.7
    Reading assignment
  • SES #: 32 · TOPICS: One-parameter groups · READINGS: Chapter 5, section 4; chapter 9, section 4 · EXERCISES: Chapter 5, 4.1a, e, 4.4; chapter 9, 5.3, and 5.5
    Reading assignment
  • SES #: 33 · TOPICS: One-parameter groups (cont.) · READINGS: Chapter 5, section 4; chapter 9 section 4 · EXERCISES: Chapter 5, 4.6; chapter 9, 5.2, 5.7, 5.10, and 7.3
    Reading assignment
  • SES #: 34 · TOPICS: The lie algebra · READINGS: Chapter 9, sections 6-7 · EXERCISES: Chapter 9, 6.1, 6.2, 6.3, and 7.7
    Reading assignment
  • SES #: 35 · TOPICS: Simple groups · READINGS: Chapter 9, section 8 · EXERCISES: Chapter 9, 8.1, and 8.5
    Reading assignment

Prerequisites & course information

Prereq: 18.100A , 18.100B , 18.100P , 18.100Q , 18.090 , or permission of instructor U (Fall) 3-0-9 units

Sources