Do subspaces have distinguished bases?
type: Item
instance of: scholarly article
title: Do subspaces have distinguished bases?
zbMATH Open document ID: 0777.13014
author: Daniel R. Farkas
author: E. L. Green
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: Let \(V\) be a vector space equipped with a (totally) well ordered basis \(B\). Writing \(0\neq x\in V\) as a linear \(B\)-combination, let \(TIP(x)\) be the largest element of \(B\) occurring (with nonzero coefficient). For \(X\subseteq V\) let \(TIP(X):=\{TIP(x):x\in X\}\). Call \(x\in X\) sharp for \(X\) if \(x=1\cdot TIP(x)+\) combination of smaller \(b\in B\) for all of which \(b\notin TIP(X)\). As main result it is proved that if \(W\) is a subspace of \(V\), then the set \(SH(W)\) of vectors sharp for \(W\) forms a basis for \(W\). In a certain sense \(SH(W)\) plays the role of an orthonormal basis. The methods used yield transparent proofs for that the restriction of diagonalizable linear transformations to invariant subspaces is diagonalizable and for the unicity of the row reduced echelon form of a matrix. The monomials of the polynomial ring \(K[x_ 1,\dots,x_ n]\) form a vector space basis that comes equipped with the natural degree lexicographic ordering. In this context the relations of the notions to reduced Gröbner basis of ideals is pointed out and the usual proof of Hilbert's basis theorem (using Dickson's lemma) is provided. Finally an application to free algebras \(K\langle x_ 1,\dots,x_ n\rangle\) is given by showing that a right noetherian monomial algebra is finitely presented [\textit{J. Lewin}, Trans. Am. Math. Soc. 188, 293-308 (1974; Zbl 0343.16002)].
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 219335
zbMATH Keywords: vector space bases
zbMATH Keywords: Gröbner basis
reviewed by: Alexander Kovačec
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072656
OpenAlex ID: W1984243116
cites work: Ordering by Divisibility in Abstract Algebras
cites work: A Matrix Representation for Associative Algebras. I
cites work: The theory of Gröbner bases
DOI: 10.1216/RMJM/1181072656
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description: scientific article; zbMATH DE number 219335