<s>[INST] Can someone explain this part of the definition of a linear combination of column vectors to me? This is what I need explained: "The system Ax=b is consistent iff b can be expressed as a linear combination, where the coefficients of the linear combination are a solution of the system."
I thought a linear combination of column vectors was something like 
$$x\begin{pmatrix}1\\2\end{pmatrix}+y\begin{pmatrix}3\\4\end{pmatrix}$$
where you have to have coefficients AND variables and multiple matrices that are added together, but a system with a single solution would have just coefficients. And any solution would have just one column matrix, not a few added together.
Also, what is a situation that would cause b to NOT be able to be expressed as a linear combination? Would that just be when the system has no solution?
Am I reading too much into this, is it just saying that the system is consistent if it has a solution like 
$$t\begin{pmatrix}3\\4\end{pmatrix}\text{, or just }\begin{pmatrix}2\\1\end{pmatrix}?$$
 [/INST] $$x\begin{pmatrix}1\\2\end{pmatrix}+y\begin{pmatrix}3\\4\end{pmatrix}=\underbrace{\begin{pmatrix}1&3\\2&4\end{pmatrix}}_A\underbrace{\begin{pmatrix}x\\y\end{pmatrix}}_{\bf x}$$
If we change this slightly to $$x\begin{pmatrix}1\\2\end{pmatrix}+y\begin{pmatrix}2\\4\end{pmatrix}=\underbrace{\begin{pmatrix}1&2\\2&4\end{pmatrix}}_A\underbrace{\begin{pmatrix}x\\y\end{pmatrix}}_{\bf x}$$
then the result will never be $\begin{pmatrix}1\\1\end{pmatrix}$, for any values of $x,y$.
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