The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.

线性子空间的格拉斯曼流形对于多种应用的数学建模非常重要，从机器学习、计算机视觉和图像处理中的问题到低秩矩阵优化问题、动态低秩分解和模型简化。通过这项主要是说明性的工作，我们的目标是以适合使用基于矩阵的算法解决上述问题的方式提供有关格拉斯曼流形几何的基本事实和公式的集合。此外，我们从用正交投影仪表示子空间的方法以及将其视为正交群的商空间时揭示了格拉斯曼几何，其中子空间被识别为（正交）基的等价类。这架起了相关研究轨道的桥梁，并允许在这两种方法之间轻松过渡。最初的贡献包括用于计算格拉斯曼上的黎曼对数图的修改算法，该算法在数值上是有利的，但也允许对切割轨迹和共轭点进行更基本、更完整的描述。我们还推导出了正交投影视角下沿测地线平行传输的公式、指数图导数的公式以及雅可比场在一点消失的公式。