Scalar by tutorteam

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Scalar

In linear algebra, real numbers are called scalars and relate to vectors in a vector
space through the operation of scalar multiplication, in which a vector can be
multiplied by a number to produce another vector.

More generally, a vector space may be defined by using any field instead of real
numbers, such as complex numbers. Then the scalars of that vector space will be
the elements of the associated field.

A scalar product operation (not to be confused with scalar multiplication) may be
defined on a vector space, allowing two vectors to be multiplied to produce a
scalar. A vector space equipped with a scalar product is called an inner product
space.

The real component of a quaternion is also called its scalar part. The term is also
sometimes used informally to mean a vector, matrix, tensor, or other usually
"compound" value that is actually reduced to a single component.

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Scalar Math is studied in linear Algebra where real number is considered as scalar
and we can also relate the term scalar math with the vectors in vector space by
using the operation known as Scalar Multiplication.

In this operation a scalar or a number is multiplied by a vector to get back another
vector.

We generally define a vector space with use of some field instead of using real
number or scalar, and such field is known as Complex Number plane or field.

The vector space or the Complex Plane has many scalar on its plane and these
scalars are the elements of the field.

In scalar math, we have a matrix called scalar matrix which is used to denote a
matrix in the form KI where k denotes a scalar and I is the identity matrix.

There is a big difference between scalar product operation and scalar
multiplication operation. Scalar product is nothing but defined as the multiplication
of two vectors to produce a scalar.

One more Point to be discussed in scalar math is that the vector space with the
scalar product is known as an inner product space.

We should know that the real part component of a quaternion is also known as its
scalar part.

The term scalar we use is sometimes used also in context with a vector or matrix
or a compound when reduced to a single component.

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For instance, when we calculate the matrix product like 1 x n matrix and an n x 1
matrix, then it is generally called as a 1 x 1 matrix and it is usually known as a
scalar.

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
scalar field K is isomorphic to a coordinate vector space where the coordinates are
elements of K. For example, every real vector space of dimension n is isomorphic
to 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.

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