The range of T, also known as the image of T, is the set of all possible outputs or values that can be obtained by applying the transformation T to the elements of the domain.
The range and null space of T are complementary subspaces that together span the entire vector space. The range is the set of all non-zero vectors that are mapped to by T, while the null space is the set of all vectors that are mapped to zero by T.
Yes, it is possible for the range of T to be larger than the null space of T. This occurs when the transformation T is not one-to-one, meaning that multiple elements in the domain may map to the same element in the range.
The range of T is directly affected by changes in the transformation matrix. As the matrix is modified, the range may expand, contract, rotate, or shift depending on the specific changes made.
Understanding the range and null space of T is important in linear algebra because it allows us to analyze and manipulate transformations in a vector space. It also provides insights into the properties of a transformation and can help in solving systems of linear equations and other mathematical problems.