About the two properties of Linear Transformation

We know that to prove a transformation is linear we need to show that

$$T(x_1, y_1)+T(x_2, y_2)=T(x_1+x_2, y_1+y_2)$$

And

$$kT(x,y) = T(kx, ky)$$

But I can’t think of a transformation which satisfies the first condition but not the second one? Does anyone know one example of this??

It is well known that there are maps $$f:\mathbb R \to \mathbb R$$ such that $$f(x+y)=f(x)+f(y)$$ but $$f$$ is not of the form $$cx$$ where $$c$$ is a constant. Such a map cannot satisfy the equation $$f(kx)=kf(x)$$ and $$T(x,y)=(f(x),f(y))$$ gives a map with the required properties.
Choose a nonzero rational number $$r$$; now choose a set $$\mathcal{B}$$ so $$\mathcal{B}\cup \{r\}$$ is a basis for $$\mathbb{R}$$ over the field $$\mathbb{Q}$$. Now define $$T(r) = 1$$ and $$T$$ is 0 on $$\mathcal{B}$$. WE can extend $$T$$ linearly to all of $$\mathbb{R}$$. It is the ugly beast you seek.