Linear combination of columns In the following question I am trying to determine if vector $b$ is a linear combination of the columns of $A$. If this is false I need to explain why, but if it is true I need to write down the linear combination.
Matrix A:
$$ \begin{bmatrix}
1& -1& 2& 1\\   
2& -3& 2& 0\\  
-1& 1& 2& 3 \\ 
-3& 2& 0& 3 \end{bmatrix}
$$  
Vector B: 
$$\begin{bmatrix} 
2  \\
3  \\
6  \\
9 
\end{bmatrix}.
$$
I know that the system $Ax = b$ has a solution if and only if the vector $b$ is a linear combination of the columns of $A$. Also, the system $Ax = 0$  has nontrivial solutions if and only if the columns of $A$ are linearly dependent.

If you perform row reduction on the augmented matrix 
$$
\begin{bmatrix}
1& -1& 2& 1&|&2\\   
2& -3& 2& 0&|&3\\  
-1& 1& 2& 3 &|&6\\ 
-3& 2& 0& 3 &|&9\end{bmatrix},
$$
you'll obtain 
$$
\begin{bmatrix}
1&0&0&-1&|&-5\\
0&1&0&0&|&-3\\
0&0&1&1&|&2\\
0&0&0&0&|&0
\end{bmatrix},
$$
which shows that the system $Ax=b$ is consistent. So, yes, $b$ is a linear combination of the columns of $A$. 
