Numerical Methods in Computational Finance. Daniel J. Duffy

Скачать книгу

Subscript j Baseline right-parenthesis x Superscript j Baseline comma alpha element-of normal double struck upper R period EndLayout"/>

Now let be the set of consisting of all polynomials of at most degree of the form:

(4.11)

      Then is a subspace of P, and it is also a subspace of .

      We say that a subset X of a vector space is said to be closed under addition if whenever , then . A subset X of a vector space is said to be closed under scalar multiplication if whenever and then .

      Theorem 4.1 A subset X of a vector space is a subspace if and only if:

      (4.12)

      An exercise: let be any r elements of a vector space . Prove that the set U of all elements of that can be written in the form forms a subspace of .

      We give an example of a subset X of defined by:

      (4.13)

      It is easily verified that X is a vector space over K, but X is not a subspace of because:

      and these two quantities are thus not the same!

4.4 LINEAR INDEPENDENCE AND BASES

      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

Скачать книгу