Heres the question:

Let be defined by T(a,b,c) = (a+b, b-2c, a+2c). Determine if the vector v=(2,1,1) exists in R(T) [the range].

Heres what I got:

The matrix corresponding to this system is:

A=

After some row operations...

A=

Then I computed the augmented matrix (A|b) and simplified to:

(A|b)=

So the rank(A)=2 and rank(A|b)=2.

By a theorem, this system is consistent ( the solution set is nonempty)

Does this mean that the vector v exists in R(T), and why?