When finding out if a vector $v_4$ exists in $Span(v_1,v_2,v_3)$ I am first building an linear system of equations that in a way that solve
$$ s*v_1 + r * v_2 + t * v_3 = v_4 $$
s,r,t are elements of the real numbers.
In my specific example my coefficient matrix solves to the following
$$ \begin{matrix} 3 & 0 & 2 & 9 \\ 0 & 3 & -1 & 3 \\ 0 & 0 & 0 & 0 \\ \end{matrix} \\ $$
Implying that this system has infinite solutions. My question is, does that imply that $v_4$ is not in that Span, or can I choose my last variable t freely such as $t = \lambda$ and continue solving the system?
Yes it means that $v_4$ is in the span, notably that means that $v_4$ can be expressed as a linear combination of two vectors, for example $v_1$ and $v_2$, thus we can fix the coefficient $t$ for $v_3$ to zero (or to any arbitrary number) and solve for $s$ and $r$.