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Vector space
Vector space
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Vector space
Definition
Wiktonary
Noun
vector
space
A type of
set
of
vector
s that satisfies a specific group of
constraint
s.
A vector space is a set of vectors which can be
linearly combined
.
Each vector space has a basis and dimension.
vector space over the field
F
A set
V
, whose elements are called "vectors", together with a binary operation + forming a
module
(
V
,+), and a set
F
^{*}
of
bilinear
unary functions
f
^{*}
:
V
→
V
, each of which corresponds to a "
scalar
" element
f
of a
field
F
, such that the composition of elements of
F
^{*}
corresponds isomorphically to multiplication of elements of
F
, and such that for any vector
v
, 1
^{*}
(
v
) =
v
.
Any field
$\backslash mathbb\{F\}$
is a one-dimensional vector space over itself.
If
$\backslash mathbb\{V\}$
is a vector space over
$\backslash mathbb\{F\}$
and
S
is any set, then
$\backslash mathbb\{V\}^S=\backslash \{f|f:S\backslash rightarrow\; \backslash mathbb\{V\}\; \backslash \}$
is a vector space over
$\backslash mathbb\{F\}$
, and
$\backslash mbox\{dim\}\; (\; \backslash mathbb\{V\}^S\; )\; =\; \backslash mbox\{card\}(S)\; \backslash ,\; \backslash mbox\{dim\}\; (\backslash mathbb\{V\})$
.
If
$\backslash mathbb\{V\}$
is a vector space over
$\backslash mathbb\{F\}$
then any closed subset of
$\backslash mathbb\{V\}$
is also a vector space over
$\backslash mathbb\{F\}$
.
The above three rules suffice to construct all vector spaces.