Given some algebra $\mathcal{A}$ and $q \in \mathbb{C}$, we say that a matrix $M \in \mathrm{Mat}(n \times n ; \mathcal{A})$ is a quantum matrix in $\mathrm{Mat}_q(n \times n)$ iff the following conditions hold for any submatrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ of $M$:
\begin{equation} \label{eq:commrels} \begin{split} ca = q \cdot ac \,\,\,,\,\,\, db = q \cdot bd \,\,\,,\,\,\, ad - da = q^{-1} \cdot cb - q \cdot bc \\ ba = q \cdot ab \,\,\,,\,\,\, dc = q \cdot cd \,\,\,,\,\,\, ad - da = q^{-1} \cdot bc - q \cdot cb \\ bc = cb \,\,\,,\,\,\, ad - da = (q^{-1}-q) bc. \end{split} \end{equation}
Note that the third line of equalities follows from the first two (if $q \neq \pm 1$). The first line can be derived by considering a 'quantum vector space' spanned by 'vectors' $\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}$ and its 'dual' spanned by 'covectors' $( x'_1 \cdots x'_n )$, subject to commutation relations $x_j x_i = q x_i x_j$ and $x'_j x'_i = q x'_i x'_j$ for $i<j$ and for which all entries commute with the entries of $M$. Considering the transformed vectors \begin{equation} \begin{pmatrix} \tilde{x}_1 \\ \vdots \\ \tilde{x}_n \end{pmatrix} = M \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \,\,\, \text{i.e.} \,\,\, \tilde{x}_i = \sum_{j} M_{i,j} x_i \quad,\quad ( \tilde{x}'_1 \cdots \tilde{x}'_n ) = ( x'_1 \cdots x'_n ) M \,\,\, \text{i.e.} \,\,\, \tilde{x}'_j = \sum_{j} x'_i M_{i,j} \end{equation} Then the first line of the commutation relations can be derived from imposing that $\tilde{x}_j \tilde{x}_i = q \tilde{x}_i \tilde{x}_j$ for $i<j$ and the second line can be derived from $\tilde{x}'_j \tilde{x}'_i = q \tilde{x}'_i \tilde{x}'_j$ for $i < j$.
Standard treatments of 'quantum linear algebra' (see e.g. Chapter 4.5 in Shahn Majid's "Foundations of Quantum Group Theory") usually work with the q-commuting 'coordinate rings' $\mathbb{C}_q[x_1,\cdots,x_n] = \mathbb{C} \langle x_1,\cdots,x_n \rangle/\{x_j x_i = q x_i x_j, i < j\}$ rather than a vector space itself. And comparisons to regular linear algebra are made for commuting coordinate rings $\mathbb{C}[x_1,\cdots,x_n]=\mathbb{C} \langle x_1,\cdots,x_n \rangle /\{x_j x_i = x_i x_j, i < j\}$.
A rough version of my question is: "Are there more invariant definitions of quantum vector spaces, quantum linear maps, quantum eigenvectors, etc that don't rely on the coordinate ring itself?".
There are some more related (possibly more pointed) questions below. Any answers or intuitions for any of these questions would be appreciated.
I was told that the standard way to relate the classical coordinate ring $\mathbb{C}[x_1,\cdots,x_n]$ to a vector space is to consider the 'prime spectrum' $\mathrm{Spec}(\mathbb{C}[x_1,\cdots,x_n])$ of $\mathbb{C}[x_1,\cdots,x_n]$, which is the set of prime ideals of the ring. In this case, the prime ideals are of the form $\langle (x_1-a_1) \cdots (x_n-a_n) \rangle$ for $a_i \in \mathbb{C}$ which are in bijection with $\mathbb{C}^n$. I heard that $\mathrm{Spec}(\mathbb{C}_q[x_1,\cdots,x_n])$ should be the 'quantum vector space' we're looking for.
This leads to the question: "Is there a direct, intrinsic vector space structure on $\mathrm{Spec}(\mathbb{C}[x_1,\cdots,x_n])$ that generalizes for $q \neq 1?$".
A (maybe vague) reverse question would be: "How much linear algebra can be recovered by just considering the coordinate ring $\mathbb{C}[x_1,\cdots,x_n]$?".
The reason I'm wondering this is motivated by a hypothetical notion: "Is there a notion of a 'quantum vector bundle' and a 'quantum flat connection' on surfaces?" (an idea that Witten mused should be related to quantum groups in his famous "Quantum Field Theory and the Jones Polynomial" paper).
