For polyhedral cones in the Euclidean space, we present its conic dimension, which is invariant under linear isomorphisms that is sensitive to the number of generators of this cone, and the related notion of conic basis. We may interpret these two notions as versions of the definitions of linear dimension and linear basis for linear subspaces in the setting of polyhedral cones. We establish a conic version of the rank-nullity theorem that, in this case, is an inequality involving the conic dimensions of both the cone and its image under a linear map. We use this conic rank-nullity inequality to establish both a decomposition and a union of conic basis, involving the lineality space of the cone. We introduce the signature of a polyhedral cone and establish some results on the injectivity of a linear map and the preservation of the signature of a polyhedral cone under linear maps. In particular, we show that a linear map that acts injectively on the linear span of a polyhedral cone preserves its signature.

对于欧几里德空间中的多面体锥，我们提出了它的圆锥维数，它在线性同构下是不变的，对这个锥体的生成元数敏感，以及相关的圆锥基概念。我们可以将这两个概念解释为多面体锥体设置中线性子空间的线性维度和线性基定义的版本。我们建立了秩零定理的圆锥版本，在这种情况下，它是一个不等式，涉及线性映射下圆锥及其图像的圆锥维度。我们使用这个圆锥秩零不等式来建立一个分解和一个圆锥基的并集，涉及圆锥的线性空间。我们介绍了多面体锥的特征，并建立了一些关于线性映射的内射性和线性映射下多面体锥的特征保持的结果。特别是，我们展示了单射作用于多面体锥的线性跨度的线性映射保留其特征。