What defines the null space of a matrix?

The null space of a matrix is defined as the set of all vectors ( \mathbf{x} ) such that when the matrix ( A ) multiplies ( \mathbf{x} ), the result is the zero vector. Mathematically, this is expressed as ( A\mathbf{x} = \mathbf{0} ). This concept is fundamental in linear algebra because it provides insight into the solutions of homogeneous linear equations. The null space is also a subspace of the vector space from which ( \mathbf{x} ) is drawn, containing all the vectors that get mapped to the zero vector under the transformation defined by the matrix.

In contrast, other options do not accurately describe the null space. The set of all eigenvalues pertains to the scalar values that characterize the behavior of the matrix under transformation but does not relate directly to the input vectors that produce the zero vector. The set of all linearly independent vectors refers to a basis or a spanning set for a space and does not capture the vectors that result in a zero product with the matrix. Lastly, the concept of the inverse of a matrix multiplied by zero only addresses what happens when you have an invertible matrix and does not relate to the fundamental definition of the null space.