Least Square method, find vector x that minimises $ ||Ax-b||_2^2$ Given Matrix A  =
| 1 0 1 | 
| 1 1 2 |
| 0 -1 -1|
and b = $[1\ \ 4\ -2]^T$
find x such that $||Ax - b||_2^2$ is minimised.
I know I have to do something along the line $A^TAx = A^Tb$
got the vector $(1/3)* [4\  7\  0] ^T$.
However the answer is 
$x = (1/3)* [4\  7\  0] ^T + \lambda*[-1\ -1 \ \ 1]^T $. I have no clue where does the $\lambda*[-1\ -1 \ \ 1]^T$ come from. Really appreciate for some help.

The vector $\begin{bmatrix} -1 \\ -1 \\1\end{bmatrix}$ is in the nullspace of $A^TA$.
So $A^TAx=A^TA\begin{bmatrix} 4/3 \\ 7/3 \\0\end{bmatrix}+\lambda A^TA
\begin{bmatrix} -1 \\ -1 \\1\end{bmatrix}=A^TA\begin{bmatrix} 4/3 \\ 7/3 \\0\end{bmatrix}+0$
