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Transcript
```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.
Basis
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
where
, ...,
are elements of
or
. 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
and
is
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
(where
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),
over
(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
(1)