## Finite Dimensional Vector SpacesAs a newly minted Ph.D., Paul Halmos came to the Institute for Advanced Study in 1938--even though he did not have a fellowship--to study among the many giants of mathematics who had recently joined the faculty. He eventually became John von Neumann's research assistant, and it was one of von Neumann's inspiring lectures that spurred Halmos to write In 1983 Halmos received the coveted Steele Prize for exposition from the American Mathematical Society for "his many graduate texts in mathematics dealing with finite dimensional vector spaces, measure theory, ergodic theory, and Hilbert space." |

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### Contents

SPACES 1 Definition of vector space | 1 |

2 Examples of vector spaces | 2 |

3 Comments on notation and terminology | 4 |

U Definition of linear dependence | 5 |

5 Characterization of linear dependence | 7 |

6 Definition and construction of bases | 8 |

7 Dimension of a vector space | 10 |

8 Isomorphism of vector spaces | 12 |

17 Direct sums | 27 |

18 Dimension of a direct sum | 29 |

19 Conjugate spaces of direct sums | 31 |

TRANSFORMATIONS 20 Definitions and examples of linear trans formations | 33 |

21 Linear transformations as a vector space | 35 |

22 Products of linear transformations | 36 |

23 Polynomials in a linear transformation | 37 |

2U Inverse of a linear transformation | 39 |

9 Linear manifolds | 14 |

10 Calculus of linear manifolds | 15 |

11 Dimension of a linear manifold | 17 |

12 Conjugate spaces | 18 |

13 Notation for linear functionals | 20 |

1U Bases in conjugate spaces | 21 |

15 Reflexivity of finite dimensional spaces | 23 |

16 Annihilators of linear manifolds | 25 |

25 Definition of matrices | 43 |

26 Isomorphism between matrices and operators | 47 |

ORTHOGONALITY | 86 |

THE CLASSICAL CANONICAL FORM | 159 |

DIRECT PRODUCTS | 170 |

HILBERT SPACE | 183 |

Bibliography | 189 |