Does this equation have a typo? (Systems Biology, singular value decomposition)
You are right this is a typo. Equation 5 is just a restatement of the definition of the dot product of two vectors:
$$\mathbf{u}_k^T\cdot \mathbf{x} = \sum_{i=1}^m u_{ki}^Tx_i$$
or written in another way
$$\mathbf{u}_k^T\cdot \mathbf{x} = u_{k1}x_1+u_{k2}x_2+\cdots+u_{km}x_m.$$

In the text Equation 5 is stating that one can find a linear combination of the metabolites described in $\mathbf{x}$ whose time evolution is described simply by a linear combination of the flux rates described in $\mathbf{v}$. The coefficients of the first linear combination are given by a row vector $\mathbf{u}_k^T$ in $U^T$. From this perspective, since there are $m$ possible metabolites ($\mathbf{x}$ is a $m\times 1$ vector), there have to be $m$ coefficients, so that the indices must range from 1 to $m$ in Equation 5. You are correct that $u_{ki}^T$ might be 0 for some $i$. But that doesn't change the fact that there should be $m$ (possibly vanishing) coefficients.