Why does the following SIS-based decision language not make sense? I'm currently reading about important lattices problems and noticed that while CVP, SVP, and LWE have decisional versions, SIS does not.  I read in the question Relation between decisional SIS and leftover hash lemma in lattices that deciding between the distributions (A, u) when $A \in Z_{q}^{n \times m}$ and $u = Ax$ for some short x, versus the case that "u" is random is unfeasible because these distributions are statistically close.

However, what keeps us from looking on this language (which seems closely related to SIS to me), and ask how hard is it to decide it?

$$L_{m,n,\beta, q} = \{(A, u) \in Z_{q}^{n \times m}\times Z_{q}^{n}\,|\, \exists x \in Z_{q}^{m}: Ax = u \wedge ||x|| \leq \beta    \}$$

This language seems well defined to me and in the case of $\beta \leq q$ it feels non-trivial too (looking on the first norm to keep the combinatorics simple - at most $(\frac{\beta}{q})^{m}$ of u's are in the language for each A...)
This is equivalent to an LWE language. 

More specifically, if A is non-singular, write it as $A = [B | C]$ with $C$ is square and invertible mod q, and set $A' = C^{-1} A = [B' | I]$. Then $C^{-1} u = A' x = B' x_1 + x_2$ where $x = (x_1,x_2)$ is small. Relabel $s=x_1$ and $e=x_2$.