On the codimension of subalgebras of the algebra of matrices over a field

TL;DR

This paper proves that any proper subalgebra of the matrix algebra over a field has dimension less than one less than the full algebra, using an elementary and accessible proof.

Contribution

It provides a simple and elementary proof of the codimension bound for subalgebras of matrix algebras over any field.

Findings

Proper subalgebras have dimension less than n^2 - 1.

The proof is elementary and accessible.

The result holds over arbitrary fields.

Abstract

In this paper we provide an elementary and easy proof that a proper subalgebra of the matrix algebra $ \mathbb{K}^{n,n }$, with $n \geq3$ and $\mathbb{K}$ an arbitrary field, has dimension strictly less than $n^2-1$.