Numerical Methods in Computational Finance. Daniel J. Duffy
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Definition 4.1 A vector
Definition 4.2 A subset X of
Definition 4.3 A set is called linearly independent if it is not linearly dependent.
The elements
Summarising, the criterion for linear independence is:
(4.15)
Definition 4.4 A basis of a vector space
Definition 4.5 The dimension n of
4.5 LINEAR TRANSFORMATIONS
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
(4.16)
We see immediately that 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
Theorem 4.2 For any given