Description: The eigenspace, in the context of linear algebra, refers to the set of all eigenvectors corresponding to a specific eigenvalue of a matrix or linear operator. This set includes not only the eigenvectors, which are those that, when multiplied by the matrix, result in a vector that is a scalar multiple of the original vector, but also the zero vector. The eigenspace is a vector subspace, meaning it satisfies the properties of closure under addition and scalar multiplication. This characteristic is fundamental in the study of linear systems, as it allows for an understanding of how vectors behave in relation to the transformation represented by the matrix. The dimension of the eigenspace, known as the geometric multiplicity of the eigenvalue, provides information about the number of linearly independent vectors that can be associated with that eigenvalue. In summary, the eigenspace is crucial for the study of matrices, understanding the structure of linear transformations, as well as for solving differential equations and problems in various fields such as physics and engineering.
