<s>[INST] Find whether vector w belongs in the span $$v_1=[1,0,1,2]$$
$$v_2 = [0,1,1,3]$$
$$v_3 = [2,1,3,7]$$
$$w = [1,2,3,4]$$
We are supposed to determine if $w$ is in $\operatorname{span}(v_1,v_2,v_3)$.
 [/INST] The row reduced form:
$$
\begin{align}
  \mathbf{A} &\to \mathbf{E}_{\mathbf{A}} \\
%
\left[
\begin{array}{ccc}
 1 & 0 & 2 \\
 0 & 1 & 1 \\
 1 & 1 & 3 \\
 2 & 3 & 7 \\
\end{array}
\right]
%
 &\to
%
\left[
\begin{array}{ccc}
 1 & 0 & 2 \\
 0 & 1 & 1 \\
 0 & 0 & 0 \\
 0 & 0 & 0 \\
\end{array}
\right]
%
\end{align}
$$
Therefore the matrix rank is $\rho = 2$.
We have two elementary columns, 1 and 2. If $w$ is in the span of the column space then we must have
$$
  \left[
\begin{array}{c}
 1 \\
 2 \\
 3 \\
 4
\end{array}
\right]
= 
\alpha
  \left[
\begin{array}{c}
 1 \\
 0 \\
 1 \\
 2
\end{array}
\right]
+
\beta 
  \left[
\begin{array}{c}
 0 \\
 1 \\
 1 \\
 3
\end{array}
\right]
$$
The first two elements reveal the answer: $w$ is not in the span.
The vector $w$ can be resolved into $\color{blue}{range}$ and $\color{red}{null}$ space components
$$
\begin{align}
  w &= 
   \color{blue}{w_{\mathcal{R}(\mathbf{A})}} + 
   \color{red}{w_{\mathcal{N}(\mathbf{A})^{*}}} \\
%
\left[
\begin{array}{c}
 1 \\
 2 \\
 3 \\
 4
\end{array}
\right]
&=
\frac{1}{17}
%
\color{blue}{\left[
\begin{array}{c}
 21 \\
 50 \\
 71 \\
 56 \\
\end{array}
\right]}
+
\frac{1}{17}
\color{red}{\left[
\begin{array}{r}
 -4 \\
 -16 \\
 -20 \\
 12 \\
\end{array}
\right]}
% 
\end{align}
$$
</s>