Which of the following statements is true about the nullspace and linear transformations?

The nullspace of a linear transformation is defined as the set of all vectors that are mapped to the zero vector. This is a fundamental concept in linear algebra that illustrates how linear transformations can behave depending on the properties of the matrix involved. When you perform a linear transformation represented by a matrix ( A ), the nullspace specifically includes all vectors ( x ) for which ( A \cdot x = 0 ). Thus, the essence of the nullspace is captured by this definition, making the statement that it consists of vectors mapping to the zero vector true.

The other choices present various misconceptions about the nullspace. For instance, while the nullspace can contain infinitely many solutions when it has a dimension greater than zero, it can also contain just the zero vector (in which case it would have a trivial solution), so it does not exclusively consist of infinite solutions. Additionally, the nullspace is not equivalent to the image of the transformation, as they represent different aspects of linear mappings: the nullspace relates to the inputs that produce the zero output, whereas the image relates to all possible outputs. Finally, vectors within the nullspace do not have to be linearly independent; it's possible for the nullspace to include dependent vectors, such as