Let us start with a formal definition of span.
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 aswhere and are two arbitrary real numbers. In other words, .
The following proposition, although elementary, is extremely important.
Proposition The linear span of a set of vectors is a linear space.
Let be the linear span of vectors . Then, is the set of all vectors that can be represented as a linear combinationTake 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.
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