# When can I write a solution set in vector form as a span?

I have a few questions about writing a solution set to a system of equations as a span.

1. My first question is just a check to make sure I am not missing something fundamental. If my solution, in vector form, is a free variable times some vector $$\vec u$$ can I write the solution as span{$$\vec u$$}? For example:

\begin{align} & x_1=9x_3\\ & x_2=-3x_3\\ & x_3=x_3\\ & x_4=0\end{align} $$\Rightarrow$$ $$\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}$$ $$=$$ $$x_3\begin{bmatrix}9\\-3\\1\\0\end{bmatrix}$$ $$\Rightarrow$$ span$$\begin{Bmatrix}9\\-3\\1\\0\end{Bmatrix}$$

1. What happens if one or more basic variables cannot be expressed in terms of free variable(s) or $$0$$. For example: would I be able to describe the following solution set as a span given that $$x_4=1$$?

\begin{align} & x_1=9x_3\\ & x_2=-3x_3\\ & x_3=x_3\\ & x_4=1\end{align} $$\Rightarrow$$ $$\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}$$ $$=$$ $$\begin{bmatrix}0\\0\\0\\1\end{bmatrix}$$ $$+$$ $$x_3\begin{bmatrix}9\\-3\\1\\0\end{bmatrix}$$ $$\Rightarrow$$ span{???}

1. Finally, Is it possible to write a solution set as a span for a non-homogeneous system of equations?

What $$\text{span}(...)$$ means is that the solution set is the subspace spanned by the vectors you put inside of the parentheses. Thus, in order for you to be able to write the solution set in the $$\text{span}(...)$$ format, the solution set has to be a subspace. However, by definition, a subspace has to contain the $$0$$ vector, which is only in the solution set of homogeneous systems of equations. Therefore, you can only write the solution in the $$\text{span}(...)$$ format for homogeneous systems: It doesn't work for non-homogeneous systems because the solution sets for non-homogeneous systems aren't subspaces.
Thus, in Part 1, your solution in terms of $$\text{span}$$ is correct, but you can't do the same thing for Part 2.