ABSTRACT
In elementary linear algebra, we learn about column vectors, matrices, and the algebraic operations on these objects: vector addition, multiplication of a vector by a scalar, matrix addition, multiplication of a matrix by a scalar, matrix multiplication, matrix inversion, matrix transpose, etc. Later in linear algebra, we study abstract vector spaces and linear maps between such spaces. This abstract setting provides a powerful tool for theoretical work. But for computational applications, it is often more convenient to work with column vectors and matrices. The goal of this chapter is to give a thorough explanation of the relation between the concrete world of column vectors and matrices on the one hand, and the abstract world of vector spaces and linear maps on the other hand. We will build a dictionary linking these two worlds, one entry at a time, which explains the exact connection between each abstract concept and its concrete counterpart. Our complete matrix-theory/linearalgebra dictionary appears in the summary for this chapter.
