A Grassmannian is a mathematical space that parameterizes all possible linear subspaces of a given dimension within a larger vector space. For example, the Grassmannian G(k, n) represents all k -dimensional subspaces of an n -dimensional vector space. These spaces are important in various fields, including algebraic geometry and topology. Grassmannians can be visualized as collections of planes or lines in higher-dimensional spaces. They have applications in areas such as projective geometry, quantum mechanics, and robotics, where understanding the relationships between different dimensions is crucial for solving complex problems.