When does adding more vectors to a spanning set potentially change its property?

A spanning set consists of vectors that can be combined through linear combinations to fill a vector space. When more vectors are added to a spanning set, the effect on the properties of that set largely depends on the characteristics of the newly added vectors.

If the new vectors added to the spanning set are linearly independent from the existing vectors, they can expand the dimensionality of the span. This means the combination of the original vectors and the new ones can cover a larger space than before, thereby altering the property of the spanning set.

Conversely, if the new vectors are linearly dependent on those that are already in the set, the spanning nature remains unchanged; these dependent vectors do not add new dimensions to the set and merely reinforce the existing combinations without contributing additional information or coverage of the vector space.

Thus, the property of the spanning set is susceptible to change when additional linearly independent vectors are included, as they can increase the dimension of the span beyond what was previously covered. This is why the assertion is valid that adding more vectors can change the spanning set's properties if the new vectors are linearly independent.