Linear Transformations and Matrices

Abstract

In order to compare different mathematical systems of the same type, it is essential to study the functions from one system to another which preserve the operations of the system. Thus in calculus we study functions f which preserve the limit operation: if $\mathop {\lim }\limits_{i \to \infty } {x_i} = x$, then $\mathop {\lim }\limits_{i \to \infty } f({x_i}) = f(x)$. These are the continuous functions. For vector spaces, we investigate functions that preserve the vector space operations of addition and scalar multiplication. These are the linear transformations. In this chapter we develop the language of linear transformations and the connection between linear transformations and matrices. Deeper results about linear transformations appear later.