How can the span of linearly dependent vectors be characterized?

The span of a set of linearly dependent vectors can indeed be characterized by the fact that it remains the same regardless of the vector included. This is because linearly dependent vectors imply that at least one of the vectors in the set can be expressed as a linear combination of the others. Therefore, removing one of the dependent vectors does not change the span, as the remaining vectors can still generate the same space.

For instance, if you have three vectors ( v_1, v_2, ) and ( v_3 ) where ( v_3 ) is a linear combination of ( v_1 ) and ( v_2 ), the span of ( {v_1, v_2, v_3} ) is the same as the span of ( {v_1, v_2} ). Hence, the span is not affected by including or excluding any of the linearly dependent vectors.

This understanding emphasizes the nature of linear dependence, where the ability to generate the same space persists even when dependent vectors are altered or omitted from the set.