Daniel J. Duffy - Numerical Methods in Computational Finance

We are now interested in finding a minimal subspace U of independent vectors containing a set X ( U contains X ) such that any vector in X can be written as a linear combination of these vectors. In this case we say that X spans U . We are particularly interested in the case , that is subsets of that span itself. Such subsets always exist; for example has this property. We take an example in n -dimensional space. The vectors:

(4.14)

span because each element can be written as a linear combination:

Furthermore, any proper subset of cannot span .

Another example is the vector space generated by polynomials of the form of Equation (4.11)generated the monomials . It is clear that any proper subset of this set spans a proper subspace of ; it cannot span itself.

Definition 4.1A vector is linearly dependent on a given subset X of if x belongs to the subspace generated by the set X .

Definition 4.2A subset X of is called a linearly dependent set if it contains at least one element that is linearly dependent on the others.

Definition 4.3A set is called linearly independent if it is not linearly dependent.

The elements as defined in Equation (4.14)form a linearly independent set in , and adjoining any other vector to this set makes it linearly dependent.

Summarising, the criterion for linear independence is:

(4.15)

Definition 4.4A basis of a vector space is any linearly independent subset of which has the property that it spans .

Definition 4.5The dimension n of (denoted by dim V ) is the supremum of the (cardinal) numbers of elements in the linearly independent subsets of .

is said to be finite-dimensional if n is finite. In this case there exist linearly independent subsets with n elements but no linearly independent subsets with elements.

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: