Linear Algebra, Linear Operator State if a given map is a linear operator from V to W. Explain your answer.
$V=R^2$, $W =R^3$, $f(a,b)=(a−b+1,−a,b)$
This is what I have thus far...
$T(u+v) = T(u) + T(v)$
$T(αv) = αT(v)$
$R^2-->R^3$
$u=a1,b1$
$v=a2,b2$
$T(u+v)=T(a1,b1,a2,b2)=T(a1+a2,b1+b2)=(a1+a2-b1-b2+1,-a1-a2,b1+b2)$
$T(u)+T(v)=(a1-b1+1,-a1,b1)+(a2-b2+1,-a2,b2)$
Anyone know if this is right so far and what are the steps to do next?

You can do it that way, although your notation needs a lot of improvement.
You need to fix your expressions for $T(u)$ and $T(v)$, keep on simplifying $T(u)+T(v)$ and check whether or not it equals $T(u+v)$.
However there is an easy short cut: if $T(0)\ne0$ then $T$ is not linear.  Check it out.
