**Symbolic Linear Algebra (326.080)**

**Summer semester 2023**

**Carsten Schneider**

Time: Tuesdays, 10:15 - 11:45

Rooms:

for March 7: K 009D for March 14: K 153C for March 21,28: S2 044 for April 18: K 001A for April 25: S2 044 for May 2: K 153C for May 9: S2 044 for May 16: K 153C for May 23: K 269D for June 6, 13: HF 9905 for June 20: K 001A for June 27: K 001A

In many applications of symbolic computation (e.g., summation, integration, solving difference/differential equations) one has to solve systems of linear equations that are not defined over floating-point numbers, but for instance over rational function fields or over principle ideal domains. In this lecture we discuss how one can generalize and/or optimize the well-known linear algebra methods in order to solve such systems. The application of these methods are illustrated by various examples.

In the first part of the lecture we revisit the theory of linear algebra in a more general setting: We do not work over fields but general rings. In this way the underlying structures of vector spaces generalize to modules. Here we require more and more ring properties (general rings with 1, commutative rings -> noetherian rings -> principle ideal rings -> principle ideal domains -> Euclidean domains -> fields) and can derive more and more intersting (algorithmic) results of the corresponding modules. In particular, we will derive the Smith normal form and illustrate it by various examples (including, e.g., the cyclic decomposition of finitely generated groups).

The second part of the lecture deals with advanced computer algebra methods to solve linear systems over a computable field. Among them we will focus on black box linear algebra. Here one gains speed ups in linear system solving provided that one can carry out matrix multiplications efficiently.

Some slides of the lecture are collected in the following file: SymbLA.pdf

The exercises appearing within the lecture are also collected in the following file: Exercises.pdf