What describes the relationship between linear transformations and vector addition?

The correct answer highlights a crucial property of linear transformations in the context of vector spaces. Linear transformations are defined by their ability to maintain the structure of vector spaces, which includes the operation of vector addition.

Specifically, a transformation ( T: V \to W ) (where ( V ) and ( W ) are vector spaces) is considered linear if, for any two vectors ( \mathbf{u} ) and ( \mathbf{v} ) in ( V ), the transformation satisfies the condition:

[ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}). ]

This condition shows that applying the transformation ( T ) to the sum of two vectors results in the same outcome as first transforming each vector individually and then adding the results. This preservation of vector addition is a fundamental characteristic that distinguishes linear transformations from other types of mappings.

In contrast, the other choices imply various types of interactions that are inconsistent with the definition of linear transformations. For example, suggesting that linear transformations do not preserve vector addition would contradict the very definition of linearity and disrupt the structure of the vector space being transformed. Understanding this foundational aspect is essential for