Which property distinguishes linear transformations from other types of mappings?

The distinguishing property of linear transformations is that they preserve both addition and scalar multiplication. This means that for any two vectors ( u ) and ( v ), and any scalar ( c ), a linear transformation ( T ) satisfies the following properties:

1. Preservation of Vector Addition: ( T(u + v) = T(u) + T(v) )

2. Preservation of Scalar Multiplication: ( T(cu) = cT(u) )

These properties ensure that the structure of the vector space is maintained through the transformation, which is fundamental in linear algebra and various applications. Other types of mappings may not maintain these linear relationships, leading to outcomes that can distort the geometric or algebraic characteristics of the space.

The other choices do not accurately capture the essence of linear transformations. Non-linearity of output would imply a departure from linearity, which is contrary to the definition. Independence from input dimensions can apply to many types of transformations but does not specifically define linearity. Randomness in output values is characteristic of certain stochastic processes but is not relevant to the concept of linear transformations at all.