子空间运算的几何代数
Geometric Algebra for Subspace Operations
摘要：
集合论关系\in、\backslash、Δ、\cap和\cup在子空间关系中有推论。几何代数是探索这些子空间运算的理想框架。关系\in、\backslash和Δ很容易被欧几里得度量的几何代数所包含。一个简短的计算表明，满足（\cap）和连接（\cup）是解决在一个投影算子表示的帮助下，一个额外的产品超出了标准的几何代数产品。结果是，即使子空间具有公共因子，也可以计算连接，并且可以在不知道连接的情况下计算相交。所有的操作都可以在任何签名（包括退化签名）中定义和计算，方法是通过线性可逆函数（LIFT到不同代数）引起的变换将问题转换为不同代数中的类似问题。新的结果，以及我们达到这些结果的技术，增加了可用于子空间计算的工具。
英文摘要：
The set theory relations \in, \backslash, Δ, \cap, and \cup have corollaries in subspace relations. Geometric Algebra is introduced as the ideal framework to explore these subspace operations. The relations \in, \backslash, and Δare easily subsumed by Geometric Algebra for Euclidean metrics. A short computation shows that the meet (\cap) and join (\cup) are resolved in a projection operator representation with the aid of one additional product beyond the standard Geometric Algebra products. The result is that the join can be computed even when the subspaces have a common factor, and the meet can be computed without knowing the join. All of the operations can be defined and computed in any signature (including degenerate signatures) by transforming the problem to an analogous problem in a different algebra through a transformation induced by a linear invertible function (a LIFT to a different algebra). The new results, as well as the techniques by which we reach them, add to the tools available for subspace computations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
