Finding vector from a given subspace Given the subspace described by those three vectors:
$W=lin([1, 0, 2, 4], [0, 1, 2, 3], [0, 1, 0, 1] ) $
Now i want to pick up any vector that is located in this subspace. How should i do this?

Definition: The linear hull/ span of a set $S$ of vectors is the set of all (finite) linear combinations of the vectors in the set.
So, by definition, any vector $v$ in the subspace $\operatorname{lin}\left( \begin{bmatrix} 1 & 0 & 2 &4 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 2 &3 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 0 &1 \end{bmatrix}\right)$ can be written as a linear combination of those $3$ vectors.  That is:
$$v=\alpha\begin{bmatrix} 1 & 0 & 2 &4 \end{bmatrix} + \beta\begin{bmatrix} 0 & 1 & 2 &3 \end{bmatrix} +\gamma\begin{bmatrix} 0 & 1 & 0 &1 \end{bmatrix}$$
where $\alpha, \beta, \gamma$ are real numbers.  So just choose your favorite three numbers, plug them into the above, and add.
