I'm trying to prove the following statement.
Let $V$ be a vector space, $W \subset V$ a subspace, and $f: V \to V/W$ the canonial quotient map sending $v \mapsto v + W$. Let $\{w_1, \ldots, w_m\}$ be a basis for $W$, and extend this to a basis $\{w_1, \ldots, w_m, v_1, \ldots, v_n\}$ of $V$. Then $\{f(v_1), \ldots, f(v_n)\}$ is a basis for $V/W$.
I'm taking for granted that $V/W$ is a vector space and has well-defined addition and scalar multiplication operations (so $f$ is linear), as well as the fact that I can extend a linearly independent set to a basis. So it remains to show that $f(v_1), \ldots, f(v_n)$ is a basis for $V/W$. Here is my attempt.
We first show that $\{f(v_1), \ldots, f(v_n)\}$ is linearly independent. Suppose that $c_1, \ldots, c_n \in F$ satisfy $$ \sum\limits_{i=1}^n c_i f(v_i) = 0 + W. $$ Then \begin{align*} 0 + W &= \sum\limits_{i=1}^n c_i f(v_i) \\ &= \sum\limits_{i=1}^n c_i (v_i + W) \\ &= \sum\limits_{i=1}^n (c_i v_i + c_i W) \\ &= \sum\limits_{i=1}^n (c_i v_i + W) \\ &= \sum\limits_{i=1}^n c_i v_i + W. \end{align*} As $0 + W = \sum\limits_{i=1}^n (c_i v_i + W)$, we have $$ \sum\limits_{i=1}^n (c_i v_i) - 0 = \sum\limits_{i=1}^n (c_i v_i) \in W. $$ As $W \subset V$, we have $\sum\limits_{i=1}^n (c_i v_i) \in V$. As $v_1, \ldots, v_n$ are linearly independent in $V$, this forces $c_1 = \ldots = c_n = 0$, so $f(v_1), \ldots, f(v_n)$ are also linearly independent. Now, let $v + W \in V/W$. Then $f(v) = v + W$. As $\{w_1, \ldots, w_m, v_1, \ldots, v_n$ is a basis for $V$ and $v \in V$, there exist $a_1, \ldots, a_m, b_1, \ldots, b_n \in F$ such that $$ v = \sum\limits_{i=1}^m a_i w_i + \sum\limits_{j=1}^n b_i v_i. $$ Applying $f$, we get \begin{align*} v + W = f(v) &= f\left(\sum\limits_{i=1}^m a_i w_i + \sum\limits_{j=1}^n b_i v_i \right) \\ &= \sum\limits_{i=1}^m a_i f(w_i) + \sum\limits_{j=1}^n b_i f(v_i) \\ &= \sum\limits_{i=1}^n a_i (w_i + W) + \sum\limits_{j=1}^n b_i (v_i + W) \\ &= W + \sum\limits_{j=1}^n b_i v_i + W \\ &= \sum\limits_{j=1}^n b_i v_i + W \\ &= \sum\limits_{j=1}^n (b_i (v_i + W)) + W \\ &= \sum\limits_{j=1}^n (b_i f(v_i)) + W, \end{align*} so $f(v_1), \ldots, f(v_n)$ spans $V/W$.
I'm not convinced that either direction of my proof are correct, particularly the proof of spanning, as my attempt at a proof is essentially saying, "it follows from linearity of $f$." How does this look?