If I have a vector subspace $W\subset \Bbb R^4$ such that $W=\langle(1,1,-2,1),(0,1,1,-1), (1,-1,-1,1)\rangle$.
If I want the parametric equations I have to consider the generic vector $(x_1, x_2, x_3, x_4) \in W$ as a linear combination of the three generator vectors and then
$$W: \begin{cases} x_1= \lambda+ \nu\\ x_2= \lambda+\mu- \nu\\ x_3= -2\lambda+ \mu - \nu\\ x_4= \lambda- \mu+ \nu\\ \end{cases} \qquad \forall \lambda, \mu, \nu \in \Bbb R .$$
My question is : before I start to do all this do I have to be certain that the three vectors are linearly independent? Can I do a set of parametric equations with some of the generators linearly dependent from the others? In my opinion I can't because the number of parameters equals to $\dim W$ that is univocal.
The whole point of parametrizations is to provide intrinsic coordinates for the range of the parametrization in an unequivocal way (in the above example $x_1, x_2, x_3, x_4$ are extrinsic coordinates, coming from the ambient space, while $\lambda, \mu, \nu$ are intrinsic coordinates, in no way connected to the ambient space).
If $\{v_1, \dots, v_r\}$ is a linearly-dependent family that generates $W$, consider $p = \alpha_1 v_1 + \dots + \alpha_r v_r$. Let us now assume that $\dim W = d < r$ and, for simplicity, $\{v_1, \dots, v_d\}$ are linearly independent, so the remaining vectors are linear combinations of these: $v_k = c_{k,1} v_1 + \dots + c_{k,d} v_d$ for $d < k \le r$. Plugging this into the formula for $p$ we get
$$p = (\alpha_1 v_1 + \dots + \alpha_d v_d) + \alpha_{d+1} (c_{d+1,1} v_1 + \dots + c_{d+1,d} v_d) + \dots + \alpha_r (c_{r,1} v_1 + \dots + c_{r,d} v_d) = \\ (\alpha_1 + \alpha_{d+1} c_{d+1,1} + \alpha_{d+1} c_{d+2,1} + \dots + \alpha_r c_{r,1})v_1 + \dots + (\alpha_d + \alpha_{d+1} c_{d+1,d} + \alpha_{d+2} c_{d+2,d} + \dots + \alpha_r c_{r,d})v_d + \\ 0 v_{d+1} + \dots 0 v_r .$$
This shows that we have found at least to sets of coordinates for the vector $p$: one is simply $(\alpha_1, \dots, \alpha_r)$, while the other is
$$\left( \alpha_1 + \sum _{k = d+1} ^r \alpha_k c_{k,1}, \dots, \alpha_d + \sum _{k = d+1} ^r \alpha_k c_{k,d}, \underbrace{0, \dots, 0} _{r-d \text{ times}} \right) .$$
This shows that in the case of linearly dependent systems of generators, parametrizations constructed as above simply defeat the purpose of a parametrization, not being able to uniquely associate intrinsic coordinates to points - and this is why we don't use them (they are not injective).