Given $A\in\mathbb{R}^{m\times n}$, $m\geq n$, compute the (economy) QR factorisation. This gives
$$
A = QR, \quad R\in\mathbb{R}^{n\times n}.
$$
Now if $\mathrm{rank}(A)<n$, the upper triangular matrix $R$ has a staircase profile with some of the "steps" of the staircase over more than one column. Select column indices $j_1,\ldots,j_k$ such that if you remove these columns from $R$, you obtain a nonsingular upper triangular matrix (you can consider it as making each step of the staircase of length 1). The columns $j_1,\ldots,j_k$ can be expressed as linear combination of the remaining columns.
Example: The red columns indicate the columns which are linear combinations of the others.
$$
\begin{bmatrix}
\times & \times & \color{red}\times & \times & \color{red}\times & \color{red}\times \\
0 & \times & \color{red}\times & \times & \color{red}\times & \color{red}\times \\
0 & 0 & \color{red}0 & \times & \color{red}\times & \color{red}\times
\end{bmatrix}
$$
Example: For the given matrix from the question, the QR factorisation is:
Q =
0 -0.4472 -0.8944
0 -0.8944 0.4472
-1.0000 0 0
R =
-1.0000 2.0000 -1.0000
0 4.4721 -2.2361
0 0 0
So one can pick the column 2 or 3 to make the matrix $R$ nonsingular and upper triangular (hence either the column 2 or 3 is a linear combination of the others).
