Heres the question:
Let 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= \left[ \begin{array}{ccc}<br />
1 & 1 & 0 \\<br />
0 & 1 &-2\\<br />
1 & 0 & 2\\<br />
\end{array}\right]<br />
After some row operations...
A= \left[ \begin{array}{ccc}<br />
1 & 0 & 2 \\<br />
0 & 1 &-2\\<br />
0 & 0 & 0\\<br />
\end{array}\right]<br />
Then I computed the augmented matrix (A|b) and simplified to:
(A|b)= \left[ \begin{array}{ccc|c}<br />
1 & 1 & 0 & 2\\<br />
0 & 1 &-2 & 1\\<br />
0 & 0 & 0 & 0\\<br />
\end{array}\right]<br />
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?