Vector Space

algebra linear-algebra

Definition

Vector Space

Let be a field. A vector space over is an Abelian group equipped with a scalar multiplication .

Elements of are called vectors; elements of are called scalars.

Subspace

Unterraum oder Teilraum

Subspace

A vector space is called subspace of vector space , denoted as , if:

The following is called subspace generated by vector , where :

Cospace

Nebenraum

Cospace

Let be a subspace of a vector space and .

Where is called cospace of at .

Basis

Definition

Basis (Vector Space)

Let be a vector space. A set is a basis of if is linearly independent and , where denotes the linear span of .

Every can be uniquely expressed as a linear combination of vectors from . The coefficients of this combination are the coordinates of with respect to .

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Basis Selection Lemma

Definition

Basis Selection Lemma

Let be a basis of a vector space and let

be a linear combination with for some . Then the exchanged set

is also a basis of . In particular, is linearly independent and , where denotes the linear span.

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Canonical Basis

Definition

Canonical Basis

Let be a -dimensional vector space. The canonical basis defined as:

where each vector has a in the -th position and elsewhere:

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Coordinate Projection

A projection maps a vector to its coordinates w.r.t. to the basis :

Due to the uniqueness of the representation as a linear combination of a basis, the coordinate is bijective and satisfies the isomorphic properties:

with and .

Thus, all arithmetic operations on can be performed in vector space . and have the same structure, hence, they are called isomorphic: