Linear span

In linear algebra, the linear span (also the tension, Span [ from English ], clamping or final called ) to a subset of a vector space over a field, the set of all linear combinations of vectors from scalars from. The linear hull forms a subspace, which is also the smallest subspace that contains.

Definition

Constructive definition

Is a vector space on a body and a portion of the vector space, then

The linear span of. The linear case, the set of all linear combinations of the finite.

In the case of a finite subset of this definition simplifies to

The linear span of the empty set is the zero vector space, that is

Because the empty sum of vectors gives by definition the zero vector.

Other definitions

Equivalent ( equal) to the structural definition of the following definitions:

The linear span of a subset of a vector space is the smallest subspace that contains the amount.
The linear span of a subset of a vector space is the intersection of all subspaces of containing.

Notation

Properties

Be two sets of subsets of the K- vector space. Then:

These three properties characterize the linear hull as a closure operator.

Next apply:

The linear span of a subset of a vector space is a subspace of.

For every subspace of a vector space.

A set of vectors is a generating set of their linear span. In particular, if a set of vectors a generating set of a subspace, so this is their linear span.

The sum of two subspaces is the linear hull of the union, ie.

In the set of subspaces of a vector space (including the total area ), you can the operation " constitutes the linear hull of the union of " two-digit shortcut introduce. The corresponding dual link is the intersection of education. With these links then forms a lattice.