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!