In the context of linear mappings, which property must hold?

The property that a linear mapping must uphold is the preservation of vector addition. This means that if you have two vectors, when you apply the linear mapping to the sum of those two vectors, the result must be the same as if you first applied the linear mapping to each vector individually and then summed the results. Mathematically, this can be expressed as:

T(u + v) = T(u) + T(v)

for any vectors u and v in the vector space. This requirement ensures that the structure of the vector space is maintained under the mapping, which is a fundamental characteristic of linear transformations.

This property is crucial because it confirms that the linear mapping behaves predictably in terms of how it interacts with vector addition, thus allowing for consistent and reliable analysis in linear algebra. Other properties, such as the mapping producing zero vectors, applying only to integers, or involving complex numbers, do not encapsulate the essence of linear mappings. They either misrepresent the requirements of linearity or impose unnecessary restrictions on the type of vectors or fields considered. Therefore, preserving vector addition is at the heart of defining a linear transformation.