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Vector Space
A vector space V is a set that is closed under finite vector addition and scalar multiplication. The basic example is ndimensional Euclidean space
, where every element is represented by a list of n real numbers, scalars are real
numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.
A basis of a vector space V is defined as a subset
span V. Consequently, if
can be uniquely written as
of vectors in V that are linearly independent and vector space
is a list of vectors in V, then these vectors form a basis if and only if every
, ...,
are elements of
. A vector space V will have many different bases, but there are always the same
number of basis vectors in each of them. The number of basis vectors in V is called the dimension of V. Every spanning list in
a vector space can be reduced to a basis of the vector space.
The span of subspace generated by vectors
Complete Basis
A set of orthogonal functions
is termed complete in the closed interval
function f(x) in the interval, the minimum square error
if, for every piecewise continuous
denotes the L2-norm with respect to a weighting function w(x)) converges to zero as n becomes infinite.
Symbolically, a set of functions is complete if
where the above integral is a Lebesgue integral.
Examples of complete orthogonal systems include
special type of system known as a complete biorthogonal system),
(which actually form a slightly more
Orthogonal functions
Two functions f(x) and g(x) are orthogonal over the interval
with weighting function w(x) if
If, in addition,
the functions f(x) and g(x) are said to be orthonormal.