Ok, I took a look at the paper now. Thomas described the DLIN assumption, which is, however, not the assumption used in the paper you are looking at.
Furthermore, what Thomas describes is the 2-DLIN assumption, which can be generalized to the $d$-DLIN assumption in a straightforward manner:
$d$-DLIN Assumption:
Given a group $G$ of prime order $p$, the tuples
$(g_1,\ldots,g_{d+1},g_1^{r_1},\ldots, g_d^{r_d},g_{d+1}^{\sum_{i=1}^d} r_i)$
and
$(g_1,\ldots,g_{d+1},g_1^{r_1},\ldots, g_d^{r_d},g_{d+1}^{r_{d+1}})$
are indistinguishable.
What you are looking at:
The paper actually uses a matrix form of the linear assumption which has been introduced here (see Appendix A) and further used in the reference 7 of the paper cited in the paper you are looking at (which is this one).
Matrix $d$-Linear Assumption:
We have a group $G$ of prime order $p$ and $g$ a generator. Let $R = \{r_i,_j\} \in Z_p^{a\times b}$, $i\in[a],j\in[b]$ be a matrix and denote by $g^R$ the matrix $\{g_{i,j}\} =
\{g^{r_{i,j}}\} \in G^{a\times b}$, $i\in[a],j\in[b]$.
Let $Rk_i(Z_p^{a\times b})$ the set of $a\times b$ matrices over $Z_p$ of rank $i$.
Then the matrix $d$-linear assumption states, that for any integers $a$ and $b$, and for any $d \leq i < j \leq \min\{a, b\}$ the tuples $(g, g^R)$ for $R\in Rk_i(Z_p^{a\times b})$ and $(g, g^R)$ for $R\in Rk_j(Z_p^{a\times b})$ are indistinguishable.
The proof that the $d$-Linear assumption implies the matrix $d$-Linear assumption can be found in the paper references above. This should be your starting point and makes more sense in the screenshots you provided.