I am currently reading Gunnar Carlsson's "Topological Pattern Recognition for Point Cloud Data", you can find it here: http://math.stanford.edu/~gunnar/actanumericathree.pdf
I have a question about this article.
Let's say $X,Y$ are finite sets, equipped with functions $\rho : X \to [0, \infty)$, and $\sigma : Y \to [0, \infty)$ respectively. The free persistence vector space on the pair $(X, \rho)$ is the persistence vector space $\lbrace W_r \rbrace$, with $W_r \subseteq V_k(X)$ equal to the k-linear span of the set $X[r] \subseteq X$ defined by $X[r] = \lbrace x \in X | \rho(x) \leq r \rbrace$. Let's call the free persistence vector spaces on $(X,\rho)$ and $(Y,\sigma)$ by $\lbrace V_k(X,\rho)_r \rbrace$ and $\lbrace V_k(Y,\sigma)_r \rbrace$ resp. For any linear transformation from $\lbrace V_k(Y,\sigma)_r \rbrace$ to $\lbrace V_k(X,\rho)_r \rbrace$, there is a matrix $A=[a_d]$ where $d$ is $xy$ (I could not write it directly due to the latex editor problem) and for any matrix $A=[a_d]$ with some properties, there is a linear transformation $f_A: \lbrace V_k(Y,\sigma)_r \rbrace \to \lbrace V_k(X,\rho)_r \rbrace$.
My question is about page 28. The notation written below made me confused
$$A \xrightarrow[]{\theta} V_k(Y,\sigma) / im(f_A)$$
Is the RHS a quotient space? If yes, $im(f_A)$ is a subspace of $V_k(X,\rho)$, not $V_k(Y,\sigma)$, how should I see this quotient space?