The span of a set of vectors, also called linear span, is the linear space formed by all the vectors that can be written as linear combinations of the vectors belonging to the given set.

Let us start with a formal definition of span.

Definition
Let
be a linear space. Let
be
vectors. The **linear span** of
,
denoted
by
is the set of all the linear combinations
that
can be obtained by arbitrarily choosing
scalars
,
...,.

A very simple example of a linear span follows.

Example Let and be column vectors defined as follows:Let be a linear combination of and with coefficients and . Then,Thus, the linear span is the set of all vectors that can be written aswhere and are two arbitrary scalars.

The following proposition, although elementary, is extremely important.

Proposition The linear span of a set of vectors is a linear space.

Proof

Let be the linear span of vectors . Then, is the set of all vectors that can be represented as linear combinationsTake two vectors and belonging to . Then, there exist coefficients and such thatThe span is a linear space if and only if, for any two coefficients and , the linear combinationalso belongs to . But,Thus, the linear combination can itself be expressed as a linear combination of the vectors with coefficients , ..., . As a consequence, it belongs to the span . In summary, we have proved that any linear combination of vectors belonging to the span also belongs to the span . This means that is a linear space.

Below you can find some exercises with explained solutions.

Define the following vectors:

Does belong to the linear span of and ?

Solution

The linear span of and is the set of all vectors that can be written as linear combinations of and with scalar coefficients and :In other words, contains all the scalar multiples of the vectorBut is not a scalar multiple of . Therefore, does not belong to .

Does the zero vectorbelong to the span of the vectors and defined above?

Solution

We have proved that the span is a linear space, and the zero vector always belongs to a linear space (by the very definition of linear space).

Please cite as:

Taboga, Marco (2021). "Linear span", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/linear-span.

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