Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 73

Mappings between vector spaces are at least as interesting as vector spaces themselves. An important property of linear transformations is that they map linearly dependent subsets into linearly dependent subsets. An interesting remark is that the set of all linear transformations between two given vector spaces is itself a vector space.

The mapping:

is called a linear transformation from to if:

(4.16)

We see immediately that the zero element in is mapped to the zero element in .

Some examples of linear transformations are:

A more general linear transformation (in fact, a vector-valued transformation) is:

Another example of a mapping is fixed, and this is called a magnification or a dilation.

Theorem 4.2 For any given :

is a necessary and sufficient condition for a mapping to be a linear transformation. From this result we can conclude that a linear transformation is determined by the images in of a basis in .