Heres the question:

Let $\displaystyle T:\mathbb{R}^3\to \mathbb{R}^3$ 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=$\displaystyle \left[ \begin{array}{ccc}

1 & 1 & 0 \\

0 & 1 &-2\\

1 & 0 & 2\\

\end{array}\right]

$

After some row operations...

A=$\displaystyle \left[ \begin{array}{ccc}

1 & 0 & 2 \\

0 & 1 &-2\\

0 & 0 & 0\\

\end{array}\right]

$

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

(A|b)=$\displaystyle \left[ \begin{array}{ccc|c}

1 & 1 & 0 & 2\\

0 & 1 &-2 & 1\\

0 & 0 & 0 & 0\\

\end{array}\right]

$

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?