How can I prove that the transformation $$ T: R^3 \to R^2 : \ T(a_1, a_2, a_3) = (a_1-a_2, 2a_3)$$ is one-to-one and/or onto?
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For one-to-one you need to show that if two points in $\mathbb R^3$ are mapped to the same point, they are the same. So let $x=(x_1,x_2,x_3)$ and similarly for $y$. Then assume $T(x)=T(y)$, insert the definition of $T$ and see if you can prove that $x_1=y_1$ and so on.
For onto, you need to show that all points in $\mathbb R^2$ can be reached. So let $x=(x_1,x_2)$ and see if you can find a $y$ such that $T(y)=x$. You should be able to find many of them. If you do this part first, it answers the other as well.