Which characteristic is NOT indicative of a linear transformation?

A linear transformation must satisfy two key properties: it must preserve vector addition and scalar multiplication. These properties ensure that the transformation is systematic and predictable.

The statement regarding an arbitrary transformation of the input vectors does not align with the characteristics of linear transformations. If a transformation is arbitrary, it may not consistently adhere to the rules of linearity. Arbitrary transformations can alter vectors in ways that do not respect their linear structure, potentially mixing operations and leading to results that do not maintain the foundation needed for linear transformations.

In contrast, scalar multiplication's effect, the preservation of combinations of vector addition, and the preservation of the origin point are all essential characteristics of linear transformations. These properties guarantee that the output of the transformation remains within the confines of linearity, allowing for predictable manipulation of vectors. Therefore, the defining nature of a linear transformation fundamentally excludes any arbitrary operations on input vectors.