Determine if the transformation T(x,y) = (2x, 2y) from R2 to R2 is linear.

Let $\vec{u} = (x_1, y_1)$ and $\vec{v} = (x_2, y_2)$ be two arbitrary vectors in $\mathbb{R}^2$.
First, calculate $T(\vec{u} + \vec{v})$:

$$\vec{u} + \vec{v} = (x_1 + x_2, y_1 + y_2)$$ $$T(\vec{u} + \vec{v}) = T(x_1 + x_2, y_1 + y_2) = (2(x_1 + x_2), 2(y_1 + y_2)) = (2x_1 + 2x_2, 2y_1 + 2y_2)$$

Next, calculate $T(\vec{u}) + T(\vec{v})$:

$$T(\vec{u}) = T(x_1, y_1) = (2x_1, 2y_1)$$ $$T(\vec{v}) = T(x_2, y_2) = (2x_2, 2y_2)$$ $$T(\vec{u}) + T(\vec{v}) = (2x_1, 2y_1) + (2x_2, 2y_2) = (2x_1 + 2x_2, 2y_1 + 2y_2)$$

Since $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$, the additivity property holds.

Let $\vec{u} = (x, y)$ be an arbitrary vector in $\mathbb{R}^2$ and $c$ be an arbitrary scalar.
First, calculate $T(c\vec{u})$:

$$c\vec{u} = (cx, cy)$$ $$T(c\vec{u}) = T(cx, cy) = (2(cx), 2(cy)) = (2cx, 2cy)$$

Next, calculate $cT(\vec{u})$:

$$T(\vec{u}) = T(x, y) = (2x, 2y)$$ $$cT(\vec{u}) = c(2x, 2y) = (c \cdot 2x, c \cdot 2y) = (2cx, 2cy)$$

Since $T(c\vec{u}) = cT(\vec{u})$, the homogeneity property holds.

Since both the additivity and homogeneity properties are satisfied, the transformation $T$ is linear.

The transformation $T(x,y) = (2x, 2y)$ is indeed $\boxed{\text{linear}}$.