Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 71

A non-empty subset X of a vector space ) is called a vector subspace of if X forms a vector space over K with the same addition and scalar multiplication as in . For example, let P be the set of polynomials in X with real coefficients, and let polynomial addition and multiplication by real numbers be defined by:

(4.10)

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!