Which of these transformations is linear?

Question

$A.$ $y1 = 10x1 - 4x2 + 8x3$

$y2 = 2x2 - 6x3$

$y3 = -7x1 - 3x2$

$B.$ $y1 = 6$

$y2 = 3$

$y3 = 7$

$C.$ $y1 = 0$

$y2 = 3x2$

$D.$ $y1 = 0$

$y2 = x1x2$

$E.$ $y1 = -8x1$

$y2 = 4x1$

$y3 = 5x1$

$F.$ $y1 = x1 + 2$

$y2 = x2$

I initially thought that only $A$ and $E$ would work as they were the only ones that mapped one $y$ to one variable of $x$. But this was incorrect, so I added $B$ as well, but this was also incorrect.

It seems like I may be misunderstanding the concept of linear transformations.

Any help or guidance would be highly appreciated!

Answer 1

Using the fact that $T(0)=0$ for linear transform, we rule out $B$ and $F$.

$C$ and $D$ are not linear because $T(cx)=cT(x)$ is violated.

For $C$, $T(1)=(0,3)$ but $T(-1)=(0,3) \ne -T(1)$.

For $D$, $T(1,1)=(0,1)$ but $T(2,2)=(0,4) \neq 2T(1,1)$.

Hence you are right that $A$ and $E$ are indeed the correct answer. Just check that they are closed under addition and scalar multiplication.