What does ker(L) represent in the context of linear transformations?

In the context of linear transformations, the notation ker(L) refers specifically to the kernel of the linear transformation L. The kernel is defined as the set of all input vectors that are mapped to the zero vector by the transformation L. In matrix terms, if L is represented by a matrix A, then the kernel is equivalent to the nullspace of A.

The nullspace of a matrix consists of all vectors x such that Ax = 0. This is a fundamental concept in linear algebra, as it provides insight into the solutions of linear equations represented by the matrix. The dimension of the nullspace is known as the nullity of the matrix, and it plays a critical role in understanding the linear independence of the columns and the overall structure of the vector space related to the transformation.

The other choices—image space, column space, and row space—represent different aspects of linear transformations and matrices. The image space relates to the outputs produced by the transformation, while the column space comprises all linear combinations of the columns of the matrix A. The row space refers to the span of the rows of the matrix. Each of these concepts has its own significance, but they do not correspond to ker(L), which specifically denotes the nullspace. Hence, the selection