Let $V$ be a vector space and $W$ a subspace. I want to find a basis for $V/W$ without giving first a basis of $W$, extending it to a basis of $V$ and taking the classes of the added vectors (the standard way to do it).
Suppose $\{v_1,\ldots,v_n\}$ is a basis for $V$. Is the set of distinct classes in $V/W$ of the vectors $v_i$ a basis for $V/W$?
I can see this is a set of generators of $V/W$, but I can't prove linear independence.
Consider $V=\Bbb{R}^2$ with the standard basis $\{e_1,e_2\}$ and $W=\{(v_1,v_2)\in V:\ v_1=v_2\}$. Then the two standard basis vectors map to two distinct elements of $V/W$, which is one-dimensional, so they cannot be linearly independent.
A more general and perhaps simpler example; extend a basis for $W$ to a basis for $V$. Then all basis vectors of $W$ are mapped to $0$ in $V/W$, so the resulting set cannot be linearly independent.