Consider the vector space $\mathbb{R}^4$ and a generic hyperplane $U = \{(a, b,c,d)\in\mathbb{R}^4: \lambda_2b + \lambda_3c + \lambda_4d = 0, \lambda_i\in\mathbb{R}_{\neq 0} \}$. The task is to find a basis of $U$.
For context I'm currently marking papers of some students in a second year university linear algebra module containing a question very similar to this, except with a specific example. I know like the back of my hand how to do this. Just take a generic vector from $U$, substitute one of the coordinates $b,c$ or $d$ in terms of the other two and after expanding the vector out the three you are left with are a basis.
However, one of the students I'm marking did the following (I'm doing it in terms of the general case I've presented here): $$ \left(\begin{matrix} a \\ b \\ c \\ d \end{matrix} \right) = \left(\begin{matrix} a \\ -\frac{\lambda_3}{\lambda_2}c - \frac{\lambda_3}{\lambda_2}d \\ -\frac{\lambda_2}{\lambda_3}b - \frac{\lambda_4}{\lambda_3}d \\ -\frac{\lambda_2}{\lambda_4}b - \frac{\lambda_3}{\lambda_4}c \end{matrix}\right) = a\left(\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix} \right) + b\left(\begin{matrix} 0 \\ 0 \\ -\frac{\lambda_2}{\lambda_3} \\ -\frac{\lambda_2}{\lambda_4} \end{matrix} \right) + c\left(\begin{matrix} 0 \\ -\frac{\lambda_3}{\lambda_2} \\ 0 \\ -\frac{\lambda_3}{\lambda_4} \end{matrix} \right) + d\left(\begin{matrix} 0 \\ -\frac{\lambda_4}{\lambda_2} \\ -\frac{\lambda_4}{\lambda_3} \\ 0 \end{matrix} \right) $$ as in, instead of just substituting one coordinate it terms of the others, they substituted all three, and then went on to show that these four vectors are linearly independent. Which as it turns out is always true, and so went on to "show" that $U = \mathbb{R}^4$, which is of course not true, since hyperplanes always have codimension $1$.
The issue is that I'm not entirely sure why this is incorrect. If a student were to ask me why this method doesn't work, I wouldn't know what to tell them. So why doesn't this work? And why are these vectors always linearly independent? What is being demonstrated when we show these vectors are linearly independent?