Find a basis of a given subspace I have this subspace $S_1 = \{(a, b, c, d) \in \mathbb{R}^4 | -a + 5b - 2c - 8d = 0\}$
And I need to find a basis in it.
The subspace represents all vectors in the form of $(5b - 2c - 8d, b, c, d)$
And thus
$\left\{
\begin{pmatrix} 5\\ 1\\ 0\\ 0 \end{pmatrix},
\begin{pmatrix} -2\\ 0\\ 1\\ 0 \end{pmatrix},
\begin{pmatrix} -8\\ 0\\ 0\\ 1 \end{pmatrix}
\right\}$
spans $S_1$, but this is not a basis since the vectors are not linearly independent. So how would I find a basis?
 A: The family of vectors that you provide contains vectors that ARE linearly independent, and so that is a basis.
However, in general, if you have a family of vectors that spans a subspace, for that family to be a basis, the number of elements in that family needs to be the same as the dimension of the subspace, or equivalently, the matrix of this family of vectors (used as columns in the matrix) must have $n$ columns and be of rank $n$ (with $n$ the dimension of the subspace).
In practice, you'd remove vectors from your spanning family until you can't remove any more without breaking the fact that the family is spanning (ie, getting back a span with lesser dimension than what you want). That's how you can ensure both the linear independence and span/generation conditions necessary/sufficient for a family of vectors to be a basis.
PS: in your case, your equation follows the standard formula for a hyperplane, where that hyperplane is normal (perpendicular) to the vector of coefficients of the equation. A hyperplane is a $(n-1)$-dimensional construct for this reason (perpendicular to a single normal vector). Here, $n = (−1, 5, −2, −8)$ and $span(n) \oplus S_1 = \Bbb R^4$. This is why you can be confident that the basis of $S_1$ has $3$ elements.
