ABSTRACT
This chapter demonstrates how to easily find a basis for the null space and the range space of a linear transformation. It explains the fundamental fact that a linear transformation can be expressed by multiplication of a vector by a matrix. However, for any particular linear transformation, the matrix will be determined by both the linear transformation and the choice of bases for the domain vector space and the range vector space. A vector space has many different bases. A fixed vector has a unique expression for a given basis, but the same vector will usually have different representations in different bases. Instead, linear transformations can be combined by function composition provided the domains and ranges are suitable.
