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?