Coordinate space

The coordinate space, standard room or standard vector space is in the mathematics of vector space of - tuples provided with components from a given body with the component-wise addition and scalar multiplication. The elements of the coordinate space is called accordingly coordinate vectors or coordinate tuple. The standard basis for the coordinate space consists of canonical unit vectors. Linear transformations between coordinate spaces can be represented by matrices. The coordinate spaces in linear algebra have a special meaning, as each finite-dimensional vector space to a coordinate space isomorphic ( structurally identical ) is.

The two - and three-dimensional real coordinate spaces often serve as models for the Euclidean plane and three-dimensional Euclidean space. In this case, its elements are interpreted either as point as well as vectors.

Definition

A body and a natural number, then the fold Cartesian product

The set of all tuples with components. For this tuple one now defines a component-wise addition by

And a component-wise multiplication by a scalar by

In this manner one obtains a vector space, which is designated as the coordinate space or standard space to the dimension across the body. Its elements are called coordinate vectors or coordinate tuple.

Representation with vectors

The coordinate vectors is often also listed as column vectors. The vector addition and scalar multiplication then corresponds to a row-wise addition of the vector components or a row- wise multiplication by a scalar:

These operations are then special cases of matrix addition and scalar multiplication of the single-column matrices.

Examples

Important examples of coordinate spaces of the choice of the real numbers as the underlying body. In the one-dimensional coordinate space the vector space operations correspond precisely to the normal addition and multiplication of numbers. In the two-dimensional real coordinate space of pairs of numbers can be interpreted as the position vectors in the Euclidean plane. The two components are then just the coordinates of the end point of a position vector in a Cartesian coordinate system. In this way corresponds to the vector addition

Clearly the addition of the associated vector arrows and the multiplication of a vector by a number

The stretching ( or compression ) of the associated vector arrow by a factor. In particular, is obtained by the vector addition or scalar multiplication again a vector in the Euclidean plane. Accordingly, the tuple of real three-dimensional coordinate space can be interpreted as positional vectors in the Euclidian space. In higher dimensions, this design works quite similarly, although the coordinate vectors of then can not be interpreted so vividly.

Properties

Neutral and inverse element

The neutral element in the coordinate space is the zero vector

Where the zero element of the body. The inverse to a vector element is then the vector

Wherein, for each of the additive inverse element in.

Legislation

The coordinate space satisfies the axioms of a vector space. In addition to the existence of a neutral and inverse element apply for coordinate vectors and scalars

• The associative law,
• The commutative,
• The mixed associative law,
• The distributive and well
• The neutrality of the one, where the unit element of the body.

These laws follow directly from the associativity, commutativity and distributivity of the addition and multiplication in the body by applying it to each component of a Koordinatentupels.

Base

The standard basis for the coordinate space consists of canonical unit vectors

Each vector can thus be a linear combination

Represent the basis vectors. The dimension of the coordinate space is given by Thus

By change of basis to the standard basis more bases in the coordinate space can be determined. It just then the column or row vectors of a matrix form a basis of coordinate space if the matrix is regular, ie, has full rank.

Isomorphism

Is now an arbitrary -dimensional vector space over the field, then is isomorphic to the corresponding coordinate space, ie

Namely, we choose a basis for, each vector has the representation

With. Each vector can thus be unambiguously represented as a tuple of coordinates. Conversely, to each such tuple of coordinates, due to the linear independence of the basis vectors exactly from a vector. Thus, the mapping

Bijective. After the picture is also linear, it provides an isomorphism between the coordinate space and the vector space dar. Since in this way every - dimensional vector space over the field to the coordinate space is isomorphic, all -dimensional vector spaces over the same body are mutually isomorphic.

This identification of finite dimensional vector spaces with the corresponding coordinate space also explains the name " standard room ". Nevertheless, it often works in linear algebra rather with abstract vector spaces instead of coordinate spaces, as you like to coordinate free in theory, that is, without a specially selected base, would argue. For specific calculations are accessed then back to the coordinate space back and expects the coordinate vectors.

Linear maps

The linear maps between two coordinate spaces correspond uniquely to matrices with entries from the body: If a matrix with rows and columns, then by the matrix-vector product is a linear map

Defined. Conversely, there is every linear mapping is a uniquely determined mapping matrix for all. The columns of result here as the images of the standard basis vectors:

The set of matrices forms of matrix addition and scalar multiplication of itself again a vector space, the die space.

Extensions

The coordinate space, for example, be extended by the following mathematical structures:

• If a real or complex coordinate space equipped with a scalar product, for example, the standard scalar product, we obtain a scalar product. As this space is completely induced by the inner product with respect to the metric, it is thereby even a Hilbert space.

• If a real or complex coordinate space provided with a vector norm, such as the Euclidean norm or other p-norm, we obtain a normed space. This room, too is then respect to the induced by the standard metric completely, ie a Banach space.

• A coordinate space is provided with a topology, such as the default topology, to obtain a topological vector space, i.e., the vector addition, and the scalar multiplication is then continuous operations.