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Matakuliah
Tahun
: D0024/Matematika Industri II
: 2008
Matematika
Pertemuan 26
Vector Space
A vector space
is a set that is closed under finite vector
addition and scalar multiplication. The basic example is dimensional Euclidean space
, where every element is
represented by a list of
real numbers, scalars are real
numbers, addition is componentwise, and scalar multiplication
is multiplication on each term separately.
For a general vector space, the scalars are members of a
field
, in which case
is called a vector space over
.
Euclidean -space
is called a real vector space,
and
is called a complex vector space.
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In order for
to be a vector space, the following conditions must
hold for all elements
and any scalars
:
1. Commutativity:
2. Associativity of vector addition:
3. Additive identity: For all
,
4. Existence
of additive inverse: For any
(
exists a2)
such that
5. Associativity of scalar multiplication:
(
3
)
6. Distributivity of scalar sums:
7. Distributivity of vector sums:
8. Scalar multiplication identity:
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, there
(
1
)
(
5
7
)
Vector Space Basis
A basis of a vector space
is defined as a subset
of
vectors in
that are linearly independent and vector space
span
. Consequently, if
is a list of vectors
in
, then these vectors form a basis if and only if
every
vector can be uniquely written as
where
, ...,
are elements of the base field. (Outside of pure mathematics,
the base field is almost always
or
, but fields of positive characteristic are
often considered in algebra, number theory, and algebraic geometry). A vector
space
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
is called the
dimension of
. Every spanning list in a vector space can be reduced to a basis
of the vector space.
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The simplest example of a basis is the standard basis in
consisting of the
coordinate axes. For example, in , the standard basis consists of two
vectors
and
. Any vector
can be written uniquely as the
linear combination
. Indeed, a vector is defined by its coordinates.
The vectors
and
are also a basis for
because any
vector
can be uniquely written as
. The above figure
shows
, which are linear combinations of the
basis
.
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When a vector space is infinite
dimensional, then a basis exists, as long
as one assumes the axiom of choice. A
subset of the basis which is linearly
independent and whose span is dense is
called a complete set, and is similar to a
basis. When
is a Hilbert space, a
complete set is called a Hilbert basis.
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