What defines the nullspace of a matrix?

The nullspace of a matrix is defined as the set of all vectors that satisfy the equation Ax = 0, where A is the matrix and x is a vector. This definition highlights that the nullspace contains precisely those vectors x that, when multiplied by the matrix A, yield the zero vector.

The nullspace plays a crucial role in linear algebra, as it provides insight into the solutions of homogeneous systems. A matrix's nullspace helps determine the number of free variables in the system, indicating the dimensionality of the solutions. The presence of a nontrivial nullspace (where the nullspace is not just the zero vector) indicates that the matrix does not have full row rank, leading to infinite solutions.

In contrast, the other options do not accurately express the definition of the nullspace. While a linear combination of vectors is related to the concept of span, it does not specifically relate to the solution of Ax = 0. The full rank of a matrix refers to the maximum number of linearly independent rows or columns, which does not directly define the nullspace. Finally, the nullspace includes all vectors satisfying Ax = 0, not exclusively dependent vectors. Thus, the specific definition focusing on the homogeneous equation clearly distinguishes the null