ABSTRACT
In this chapter the general notion of a linear transformation is presented. In Section 4.1 the definition of a linear transformation is introduced and many exam- ples are given in the context of the four classic vector spaces. Methods are given to prove or disprove that a particular map is a linear transformation. In Section4.2 two special subspaces are defined related to a linear transformation: Kernel and image. These subspaces are computed and their dimensions are determined. In Section 4.3 an important connection is made between linear transformations and matrices. In Section 4.4 the inverse linear transformation is discussed and matrix representation is used to compute it. The notion of isomorphism is discussed how two seemingly different vector spaces are essentially the same. In Section 4.5 different matrix repre- sentation of a particular linear transformation are shown to be related by similarity. In Section 4.6 the reader will learn how to compute eigenvalues, eigenvectors and eigenspaces as well as diagonalize a linear transformation or matrix. The final sections of this chapter are for advanced learners. Section 4.7 gives an axiomatic treatment of the determinant. Section 4.8 introduces quotient vector spaces. Section 4.9 introduces the dual vector space and the transpose of a linear transformation.
