 StatLect

# Linear span

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. ## Definition

Definition Let be vectors. Their linear span 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 as where and are two arbitrary real numbers. In other words, .

## A linear span is a linear space

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 a linear combination Take two vectors and belonging to . Then, there exist coefficients and such that The span is a linear space if and only if for any two coefficients and the linear combination also 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.

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