Scalar (mathematics)

A scalar is a mathematical quantity that is characterized solely by specifying a numerical value ( in physics, where appropriate, with unit).

In the mathematical subfield of linear algebra scalar denotes an element of the body of a vector space, so usually a real number. In contrast, the elements of a vector space are called vectors. According to the basic body is also called Skalarkörper. The multiplication of a vector by a scalar is the scalar multiplication or scaling. The resulting vector is called a scalar multiple of.

In contrast to the scalar, the scalar product is a bilinear form whose value is a scalar.

Scalars in physics

In physics, find scalars application, the physical quantities that are independent of direction. Examples of physical scalars are the mass of a body, its temperature, its energy and its distance from another body ( as an absolute value of the difference of the position vectors ). However, if a direction is required, as in force or speed for the adequate description of the size, it is called in the physics of a vector or tensor.

More specifically, a scalar in physics is a size that does not change under rotations, in relativistic physics and Lorentz transformations. Therefore they are also called Lorentz scalar.

The velocity of a particle has the direction in which the particle is moving. Since the direction changes during turns, the speed is not a scalar, but a vector. However, the magnitude of the velocity does not change upon rotation, and is a scalar.

If a is a scalar quantity, dependent on the considered group of transformations. Thus, energy is a scalar with respect to rotations, but in relativity theory component of a four-vector.

A subset of the scalars are the pseudoscalars, which reverse the sign under a plane reflection.

Extensions and demarcation of similar terms

• Square matrices ( regarded as a linear mapping of a vector space to itself) correspond to a multiplication of each vector with a fixed scalar, the property is attributed to scalar. You are diagonal matrices whose entries on the diagonal are all equal.

• In a module over a ring, the multiplication of a modular element with an element is called scalar multiplication of the base ring. The term scalar for the elements of the base ring in this case is only partially in use.