## Abstract

We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix (as well as other problems) are complete under logspace reductions. As an important part of presenting this classification, we show that the "exact counting logspace hierarchy" collapses to near the bottom level. We review the definition of this hierarchy below. We further show that this class is closed under NC^{1}-reducibility, and that it consists of exactly those languages that have logspace uniform span programs (introduced by Karchmer and Wigderson) over the rationals. In addition, we contrast the complexity of these problems with the complexity of determining if a system of linear equations has an integer solution.

Original language | English (US) |
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Pages (from-to) | 99-126 |

Number of pages | 28 |

Journal | Computational Complexity |

Volume | 8 |

Issue number | 2 |

DOIs | |

State | Published - 1999 |

Externally published | Yes |

## Keywords

- #L
- Linear algebra
- Logspace counting
- Probabilistic logspace
- Rank

## ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics(all)
- Computational Theory and Mathematics
- Computational Mathematics