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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??

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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.

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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.

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