<s>[INST] Finding the linear combinations of two vectors I am studying for my finals and I'm trying to answer the following question:

Consider the following two vectors in $\mathbb{R}^3$: $a=(1,2,3)$ and $b=(2,3,1)$. Decide
  whether it is possible to  express the vector $c=(2,4,5)$ as a linear
  combination of $a$ and $b$.

I have used the following row operations to find what $C_1$,$C_2$ and $C_3$ are equal to:
\begin{align*}
\left[ \begin{array}{ccc|c} 1 & 2 &2 & 2\\
2 & 3 & 4 & 4 \\
3 & 1 & 5 & 5 \end{array}\right]
&\overset{R_2 = 2R_1 -R_1}{\Longrightarrow}
\left[ \begin{array}{ccc|c} 
1 & 2 &2 &2 \\
0 & 1 & 0 & 0 \\
3 & 1 & 5 & 5 
\end{array}\right]
\overset{R_3=3R_1-R_3}{\Longrightarrow} 
\left[\begin{array}{ccc|c}
1 &2 & 2 & 2 \\
0 &1 & 0 & 0 \\
0 & 5 &1 & 1 
\end{array} \right]
\\
&\overset{R_3=5R_2+R_3}{\Longrightarrow}
\left[\begin{array}{ccc|c} 
1 & 2 & 2 & 2 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 
\end{array}\right]
\end{align*}
Is the following argument correct?

By looking at the last row of the matrix, we see that it says $0 = 1$, which is impossible, and the system of equations therefore has no solutions. This means that we can not find the values for $C_1$ and $C_2$, and so $c$ can not be written as a linear combination of $a$ and $b$. 

Thanks in advance!
 [/INST] $c=m\cdot a+n \cdot b$
Hence :
$(2,4,5)=m\cdot(1,2,3)+n\cdot(2,3,1)$
So we have following system of equations :
$\begin{cases}
 m+2n=2 \\
 2m+3n=4 \\
 3m+n=5
\end{cases}$
which has no solution ,therefore you cannot express $c$ as linear combination of $a$ and $b$ .
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