$T:R^3 \to R^2$, where $T(x,y,z)=(x+y+z, x-y-z)$
So I ask myself can every point in $R^2$ be found with the inputs from $R^3$.
So the result in $R^2$ could be seen as this system of linear equations:
$$\begin{array}{c}
x+y+z = b_1 \\
x-y-z = b_2
\end{array}
$$
and this system has more unknowns than equations so it has infinite solutions I believe, so this is surjective. But i'm also thinking that it cannot be injective because $\text{dim}(R^3) \gt \text{dim}(R^2)$.
Would this be correct?
Yes, since the system always has a solution: $x = \dfrac{b_1+b_2}{2}$. And you can let $z = 0$, then $y = \dfrac{b_1-b_2}{2}$.