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?

Yes, that's a workable aproach. You will have to continue the last line and add the vectors, and you'll see that the first term doesn't really match (you'll get $T(u)+T(v) = (a_1+a_2-b_1-b_2+2, ...) \ne (a_1+a_2-b_1-b_2+1, ...)$).
You've got a couple of errors in the last line though. When things gets messy it's important to be careful to avoid such mistakes.
