I'm quickly reviewing the concept of quotient space.
Suppose $V$ is a vector space of dimension $m$ and $W$ is a vector subspace of $V$. Then the quotient vector space is defined as $V/W$, which is a set of equivalence classes, $x ~ y$ iff $x - y \in W$, elements of $V/W$ are the cosets $[x] = x + W$.
I'm trying to find a basis an the dimension of such vector space (which I'm not proving here, but I can easily prove the axioms).
For the basis suppose $x_1, \ldots, x_{m - n}$ is a basis for $W^{\perp}$ and $x_{m - n + 1}, \ldots x_{m}$ is a basis for $W$, then
$$ [\alpha_1 x_1 + \ldots \alpha_{m-n} x_{m-n} + \alpha_{m-n+1} x_{m-n+1} + \ldots + \alpha_m x_m] = \\ \alpha_1 x_1 + \ldots \alpha_{m-n} x_{m-n} + \alpha_{m-n+1} x_{m-n+1} + \ldots + \alpha_m x_m + W = \\ (\alpha_1 x_1 + \ldots \alpha_{m-n} x_{m-n}) + (\alpha_{m-n+1} x_{m-n+1} + \ldots + \alpha_m x_m) + W $$
we specificaly have $(\alpha_{m-n+1} x_{m-n+1} + \ldots + \alpha_m x_m) + W = W$ hence $$ [\alpha_1 x_1 + \ldots \alpha_{m-n} x_{m-n} + \alpha_{m-n+1} x_{m-n+1} + \ldots + \alpha_m x_m] = \alpha_1 [ x_1 ] + \ldots + \alpha_{m-n} [ x_{m-n} ] $$
Hence we have $\left\{ [x_1],\ldots,[x_{m-n}] \right\}$ are generators for $V/W$, to show linear independence is relatively straightforward and I'll skip it. This proves that $V/W$ is a vector space of dimension $m-n$.
The map
$$ f : [\alpha_1 x_1 + \ldots + \alpha_{m-n} x_{m-n} ] \to (\alpha_1,\ldots,\alpha_{m-n}) $$
Is linear isomorphism (or homeomorphism? always confused with the difference) this implies we can identify $V/W$ with $\mathbb{R}^{m-n}$.
Is all of the stated above correct?