Final answer:
The linearity of a transformation can be proven by showing that it satisfies the properties of linearity: preservation of addition and scalar multiplication.
Step-by-step explanation:
The linearity of a transformation can be proven by showing that the transformation satisfies the properties of linearity. A transformation is linear if it preserves addition and scalar multiplication. In other words, if T is a linear transformation, it should satisfy the following conditions:
- T (u + v) = T(u) + T(v)
- T(cv) = CT(v)
To prove linearity, you can choose two vectors u and v, and then apply the transformation T to both vectors separately, and also apply the transformation to the sum of the vectors and a scalar multiple of one of the vectors. If the transformation satisfies the conditions above for all vectors u and v, then it is linear.