Do subspaces have distinguished bases?
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)].
scientific article; zbMATH DE number 219335