Scalar multiplication

The scalar multiplication, also known as S- multiplication or scalar multiplication is an external binary operation between a scalar and a vector, which is required in the definition of vector spaces. The scalars are the elements of the body, on the vector space is defined. Also, the analog link in modules is called scalar multiplication.

The result of a scalar is an appropriately scaled vector. In the illustrative case extended Euclidean vector spaces or the scalar shortens the length of the vector by the specified factor. The scalars for the negative direction of the vector is also reversed. A special form of such a scale is the normalization. Here, a vector is ( generally his standard) multiplied by the inverse of its length, yielding a unit vector with length ( or norm ) one obtains.

• 3.1 coordinate vectors
• 3.2 matrices
• 3.3 polynomials
• 3.4 Functions

Definition

Is a vector space over the field, then the scalar multiplication is a binary operation

Which is mixed by definition the vector space associative and distributive, ie for all vectors and all scalars satisfies the following properties:

In addition, the neutrality of the identity of the body shall:

• .

Herein, the vector addition as well as in each case, and the addition and the multiplication in the body. Often the mark is used for both the vector addition as well as for the addition of the body plus and both the scalar multiplication, and for the body multiplication. This convention is also followed due to the readability, in the remainder of this article. The multiplication symbol is often omitted and we write briefly held and held.

Properties

Neutrality

Identifies the zero element of the field and the zero vector of the vector space, then for all vectors

Because it applies to the second distributive law

And therefore the zero vector must be. Accordingly, valid for all scalars

Because it applies to the first distributive

And hence the zero vector must be here. Overall, one obtains

As follows either or, then, the multiplicative inverse element is.

Inverse

Now called the additive inverse element to the identity element and the inverse vector, then applies

Because with the neutrality of the one obtained

And thus to the inverse vector. Is now generally the additive inverse element, then applies

Because with obtained by the mixed associative law

And the commutativity of multiplication of two scalars

Examples

Coordinate vectors

If the coordinate space and a coordinate vector, the multiplication is componentwise defined as follows by a scalar:

Accordingly, in the scalar multiplication, each component of the vector is multiplied by a scalar. In three-dimensional space eudklidischen are obtained for example

Matrices

If the die space, and a matrix, the multiplication is also components as defined by a scalar:

Thus, each entry in the scalar multiplication of the matrix multiplied by the scalar again. For example, one obtains for a real matrix

Polynomials

Is the vector space of polynomials with coefficients in the variable of a body, the multiplication of a polynomial by a scalar is defined in turn as components:

For example, yields the scalar multiplication of the real polynomial function with the number of the polynomial

Functions

Is a linear function space, and a function of a non- empty set in a vector space, the result of the scalar multiplication of such a function is defined with a scalar function as the

For example, considering the vector space of linear real-valued functions of the form, is then obtained by scalar multiplication by a real number, the function

By scalar multiplication therefore each function value is scaled by a factor.