Numerical Methods in Computational Finance - Daniel J. Duffy

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alt="upper V left-parenthesis upper K right-parenthesis"/> itself. Such subsets always exist; for example upper X equals upper V left-parenthesis upper K right-parenthesis has this property. We take an example in n-dimensional space. The vectors:

normal lamda 1 e 1 plus ellipsis plus normal lamda Subscript n Baseline e Subscript n Baseline period

      Furthermore, any proper subset of StartSet e 1 comma ellipsis comma e Subscript n Baseline EndSet cannot span upper K Superscript n.

      Definition 4.1 A vector x element-of upper V left-parenthesis upper K right-parenthesis is linearly dependent on a given subset X of upper V left-parenthesis upper K right-parenthesis if x belongs to the subspace generated by the set X.

      Definition 4.2 A subset X of upper V left-parenthesis upper K right-parenthesis is called a linearly dependent set if it contains at least one element that is linearly dependent on the others.

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

      Summarising, the criterion for linear independence is:

      (4.15)StartLayout 1st Row 1st Column Blank 2nd Column The s e t x equals StartSet x 1 comma ellipsis comma x Subscript n Baseline EndSet is linearly independent if normal lamda 1 x 1 plus ellipsis plus normal lamda 1 x Subscript n Baseline equals 0 2nd Row 1st Column Blank 2nd Column implies normal lamda 1 equals normal lamda 2 equals ellipsis equals normal lamda Subscript n Baseline equals 0 period EndLayout

      Definition 4.4 A basis of a vector space upper V left-parenthesis upper K right-parenthesis is any linearly independent subset of upper V left-parenthesis upper K right-parenthesis which has the property that it spans upper V left-parenthesis upper K right-parenthesis.

      Definition 4.5 The dimension n of upper V left-parenthesis upper K right-parenthesis (denoted by dim V) is the supremum of the (cardinal) numbers of elements in the linearly independent subsets of upper V left-parenthesis upper K right-parenthesis.

      upper V left-parenthesis upper K right-parenthesis 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 left-parenthesis n plus 1 right-parenthesis 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:

upper T colon upper V left-parenthesis upper K right-parenthesis right-arrow upper W left-parenthesis upper K right-parenthesis

      is called a linear transformation from upper V left-parenthesis upper K right-parenthesis to upper W left-parenthesis upper K right-parenthesis if:

      (4.16)StartLayout 1st Row 1st Column Blank 2nd Column upper L Baseline 1 colon upper T left-parenthesis x 1 plus x 2 right-parenthesis equals italic upper T x 1 plus italic upper T x 2 comma for-all x 1 comma x 2 element-of upper V left-parenthesis upper K right-parenthesis 2nd Row 1st Column Blank 2nd Column upper L Baseline 2 colon upper T left-parenthesis alpha x right-parenthesis equals alpha italic upper T x comma for-all alpha element-of upper K comma for-all x element-of upper V left-parenthesis upper K right-parenthesis period EndLayout

      Some examples of linear transformations are:

StartLayout 1st Row 1st Column Blank 2nd Column italic upper T x equals 0 for-all x element-of upper V left-parenthesis upper K right-parenthesis left-parenthesis zero transformation right-parenthesis period 2nd Row 1st Column Blank 2nd Column italic upper T x equals x for-all x element-of upper V left-parenthesis upper K right-parenthesis left-parenthesis identity transformation <p style= Скачать книгу