I'm trying to find a base for a vector space that's given as a set with certain traits. Take this example:
Let $V$ be an $\mathbb{R}$-vector space with $$ V := \left\{ (a, b, c, d) \in \mathbb{R}^4 : a + 3b + 2d = 0, 2a + b +c = 0 \right\} $$ and regular vector addition and scalar multiplication.
Obviously, the constraint is that the given equation system is fulfilled: $$ \left. \begin{matrix} 1 & 3 & 0 & 2 \\ 2 & 1 & 1 & 0 \\ \end{matrix} \right| \begin{matrix} 0 \\ 0 \end{matrix} $$
When solved, that means $$ a = \frac{2\lambda - 3\mu}{5},\quad b= -\frac{4\lambda - \mu}{5},\quad c=\mu,\quad d = \lambda,\quad \lambda,\mu \in \mathbb{R} $$
Of course, I could determine a few random vectors based on that constraint, hope that they'll form a generator and determine a basis based on that. But I imagine it would only make proving that the base is indeed a base harder than necessary.
There must be a better, more straightforward way, isn't there?
Don't take random vactors. Take the vector that corresponds to $\mu=1$ and $\lambda=0$ (you will get $\left(-\frac45,\frac15,1,0\right)$) and the one that corresponds to $\mu=0$ and $\lambda=1$. Then you will get two vectors which are necessarily linearly independent.