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 of lines in three-dimensional space consists of all possible lines that can be drawn through the origin. This concept is essential in various fields, including geometry and algebra. Grassmannians can be visualized as collections of points, where each point represents a specific subspace. They are often denoted as Gr(k, n), where k is the dimension of the subspace and n is the dimension of the larger space. This structure helps mathematicians study properties of linear transformations and vector spaces.