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vectorspace

A vector space, also known as a linear space, is a set V equipped with two operations: vector addition and scalar multiplication, defined with respect to a field F. The elements of V are called vectors, and the scalars come from F.

These operations satisfy a collection of axioms: closure under addition and scalar multiplication; associativity and commutativity

Examples include R^n with standard coordinatewise addition and scalar multiplication, the space of polynomials over F,

Linear transformations are maps T: V -> W that preserve the operations: T(a v + b w) = a

Vector spaces form the foundational framework for solving linear equations, performing coordinate geometry, and modeling many

of
addition;
an
additive
identity
(the
zero
vector);
additive
inverses;
compatibility
of
scalar
multiplication
with
field
multiplication;
identity
scalar
acting
as
1;
and
distributive
properties
of
scalar
multiplication
over
vector
and
field
addition.
When
F
is
the
real
numbers,
V
is
a
real
vector
space;
when
F
is
the
complex
numbers,
it
is
a
complex
vector
space.
the
space
of
all
continuous
functions
from
a
set
to
F,
and
the
set
of
m-by-n
matrices
over
F
under
ordinary
addition
and
scalar
multiplication.
A
subspace
is
a
subset
of
V
that
is
closed
under
these
operations
and
contains
the
zero
vector.
A
basis
is
a
set
of
vectors
that
is
linearly
independent
and
spans
V,
and
the
dimension
of
V
is
the
size
of
a
basis.
T(v)
+
b
T(w).
Matrices
provide
a
concrete
representation
of
linear
transformations
once
bases
are
chosen.
The
dual
space
V*
consists
of
all
linear
functionals
from
V
to
F.
systems
across
science,
engineering,
and
mathematics.