Keane, Patrick Gerard
An algebra over a field (simply “algebra” for short) is an algebraic structure consisting of a vector space augmented with a vector multiplication operation obeying suitable axioms. Algebras are well behaved and have notions of dimension, basis, subalgebras, algebra ideals, algebra homomorphisms, and quotient algebras largely analogous to those of vector spaces or rings. Algebras occur often in mathematics, for example the set of all n×n matrices valued in a field k form an algebra M_n(k). We investigate which integers occur as dimensions of subalgebras of M_n(k). We give a description of the dimensions of simple, nilpotent, and semisimple matrix subalgebras along with several sequences that represent various properties of matrix subalgebras.
Worcester Polytechnic Institute
Major Qualifying Project
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