Final answer:
A transformation T is linear if it satisfies the conditions of being additive and homogeneous of degree 1, meaning T(u + v) = T(u) + T(v) and T(cu) = cT(u), respectively, which corresponds to option A.
Step-by-step explanation:
A transformation T is linear if it satisfies two main conditions: it must be additive and homogeneous of degree 1. The additive property means that T(u + v) should equal T(u) + T(v), whereas the homogeneous property implies that T(cu) should be equal to cT(u), where u and v are vectors and c is a scalar.
The correct answer to the question is A. T(u + v) = T(u) + T(v) and T(cu) = cT(u). These conditions ensure that T preserves vector addition and scalar multiplication. Therefore, the two defining properties of a linear transformation are provided in option A.
Brief mentions of velocity transformations, vector algebra, and kinematic equations can help illustrate the importance of thinking in terms of linear transformations within physics as well as mathematics. However, they are not central to answering the specific question posed by the student.