Verify that T is a linear transformation Define
  $$ T\pmatrix  { \begin {bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} }=\begin {bmatrix} x_1+x_3 \\ x_2+x_3 \end{bmatrix}$$
  Verify that T is a linear transformation


I know that this is not the best format, and I understand if you are frustrated, but if you can resist, please don't post a comment just telling me that I should write my questions better.

I have just a few questions: Since there are three inputs, shouldn't there be three outputs as well? Is there a zero in there somewhere? Like, should I read it like this: T(x1, x2, x3) = (x1 + x2, 0, x2 + x3), where the zero is in the middle? How is it supposed to be read?
A function is a linear transformation if

$T(\mathbf x + \mathbf y) = T(\mathbf x) + T(\mathbf y)$ and $T(a\mathbf x) = aT(\mathbf x)$

Does this satisfy those two rules?

As for your other question, why can't a function have a different codomain from its domain?

Isn't $f(x,y) = x+y$ a valid function (and a linear map).  

Going the other way, $f(t) = (t, 2t, 3t)$ is also a linear map.
$$T (1,2,3)=(4,5)$$

the image of a triplet is  a couple.