A transformation $T$ is said to be linear if and only if it satisfies the following property:

$T(\alpha x + \beta y) = \alpha T(x) + \beta T(y)$ where $\alpha, \beta \in \mathbb{R}$ (real numbers), and $x, y$ are vectors in the domain of $T$.

This property is known as the linearity condition and consists of two parts:

1. Additivity: $T(x + y) = T(x) + T(y)$
2. Homogeneity (or scalar multiplication): $T(\alpha x) = \alpha T(x)$

Therefore, the correct answer is:

• T(αx+βy) = αT(x) + βT(y), α,β ∈ R

Would you like more details or have any questions?

Here are five questions related to this:

1. What is the difference between a linear and non-linear transformation?
2. How does the concept of linearity apply to matrix transformations?
3. Can a linear transformation be both one-to-one and onto?
4. How do linear transformations relate to vector spaces?
5. What is the geometric interpretation of a linear transformation in 2D or 3D space?

Tip: When checking for linearity, always test for both additivity and homogeneity.