  The randomized singular value decomposition proposed in [12] has certainly
become one of the most well-established randomization-based algorithms in
numerical linear algebra. The key ingredient of the entire procedure is the
computation of a subspace which is close to the column space of the target
matrix $\mathbf{A}$ up to a certain probabilistic confidence. In this paper we
propose a modification to the standard randomized SVD procedure which leads, in
general, to better approximations to $\text{Range}(\mathbf{A})$ at the same
computational cost. To this end, we explicitly construct information from the
row space of $\mathbf{A}$ enhancing the quality of our approximation. We also
observe that very few pieces of information from $\text{Range}(\mathbf{A}^T)$
are indeed necessary. We thus design a variant of our algorithm equipped with a
subsampling step which largely increases the efficiency of our procedure while
attaining competitive accuracy records. Our findings are supported by both
theoretical analysis and numerical results.
