Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 84

We are almost finished with our introduction to vector spaces and linear transformations. We now discuss how matrices arise and their relationship with the current topics.

We discuss the notion of a matrix for a linear transformation. To this end, consider the linear transformation where V and W are finite-dimensional vector spaces, and let and be bases in V and W, respectively. Then:

(5.10)

for some scalars . We can represent these scalars in rectangular form which we call a matrix:

(5.11)

In this case we speak of a square matrix when otherwise, it is called a rectangular matrix. In short, each linear transformation determines a unique matrix with respect to the basis functions. Conversely, every such matrix determines a unique linear transformation from V to W defined by Equation (5.10) and the following mapping for a general vector :

(5.12)