used in linear algebra, as opposed to
vectors. This image shows a
Euclidean vector. Its coordinates
x and
y are scalars, as is its length, but
v is not a scalar.
Scalars of vector spaces A
vector space is defined as a set of vectors (additive
abelian group), a set of scalars (
field), and a scalar multiplication operation that takes a scalar
k and a vector
v to form another vector
kv. For example, in a
coordinate space, the scalar multiplication k(v_1, v_2, \dots, v_n) yields (k v_1, k v_2, \dots, k v_n). In a (linear)
function space, is the function . The scalars can be taken from any field, including the
rational,
algebraic, real, and complex numbers, as well as
finite fields.
Scalars as vector components According to a fundamental theorem of linear algebra, every vector space has a
basis. It follows that every vector space over a field
K is
isomorphic to the corresponding
coordinate vector space where each
coordinate consists of elements of
K (E.g., coordinates (
a1,
a2, ...,
an) where
ai ∈
K and
n is the dimension of the vector space in consideration.). For example, every real vector space of
dimension n is isomorphic to the
n-dimensional real space
Rn.
Scalars in normed vector spaces Alternatively, a vector space
V can be equipped with a
norm function that assigns to every vector
v in
V a scalar ||
v||. By definition, multiplying
v by a scalar
k also multiplies its norm by |
k|. If ||
v|| is interpreted as the
length of
v, this operation can be described as
scaling the length of
v by
k. A vector space equipped with a norm is called a
normed vector space (or
normed linear space). The norm is usually defined to be an element of
V's scalar field
K, which restricts the latter to fields that support the notion of sign. Moreover, if
V has dimension 2 or more,
K must be closed under square root, as well as the four arithmetic operations; thus the rational numbers
Q are excluded, but the
surd field is acceptable. For this reason, not every scalar product space is a normed vector space.
Scalars in modules When the requirement that the set of scalars form a field is relaxed so that it need only form a
ring (so that, for example, the division of scalars need not be defined, or the scalars need not be
commutative), the resulting more general algebraic structure is called a
module. In this case the "scalars" may be complicated objects. For instance, if
R is a ring, the vectors of the product space
Rn can be made into a module with the
n ×
n matrices with entries from
R as the scalars. Another example comes from
manifold theory, where the space of
sections of the
tangent bundle forms a module over the
algebra of real functions on the manifold.
Scaling transformation The scalar multiplication of vector spaces and modules is a special case of
scaling, a kind of
linear transformation. ==See also==