Let $u,v,w \in \mathbb{R}^4$ as :
$u= \begin{pmatrix} 2\\ 1 \\0\\1 \end{pmatrix}$ $v= \begin{pmatrix} 1\\ 3 \\1\\-1 \end{pmatrix}$ $u= \begin{pmatrix} -1\\ 0 \\1\\2 \end{pmatrix}$
I have to find a system of equation of $Vect(u,v,w)$ but I always fail to find it...
How many equation must it have?
How to find it the fastest way?
Guessing that the third vector you denoted is actually $w$, we find that $x \in \text{vect}(u,v,w)$ if and only if $x = \lambda_1u + \lambda_2v + \lambda_3w$ where $\lambda_i \in \mathbb{R}$. This gives you your system of equations.
To find the number of equations, you can define a matrix and look at its rank: the matrix is the following: \begin{equation} \begin{pmatrix} 2 & 1 & 0 & 1 \\ 1 & 3 & 1 & -1\\ -1 & 0 & 1 & 2 \end{pmatrix}. \end{equation} This matrix follows from $x = \lambda_1u + \lambda_2v + \lambda_3w$. If you use Gaussian Elimination, you should find that this matrix is equivalent to $$ \begin{pmatrix} 1 & 0 & 0 & 1.5\\ 0 & 1 & 0 & -2\\ 0 & 0 & 1 & 3.5 \end{pmatrix}$$ so it has rank 3.