  We prove an estimate on the smallest singular value of a multiplicatively and
additively deformed random rectangular matrix. Suppose $n\le N \le M \le
\Lambda N$ for some constant $\Lambda \ge 1$. Let $X$ be an $M\times n$ random
matrix with independent and identically distributed entries, which have zero
mean, unit variance and arbitrarily high moments. Let $T$ be an $N\times M$
deterministic matrix with comparable singular values $c\le s_{N}(T) \le
s_{1}(T) \le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\times n$
deterministic matrix with $\|A\|=O(\sqrt{N})$. Then we prove that for any
$\epsilon>0$, the smallest singular value of $TX-A$ is larger than
$N^{-\epsilon}(\sqrt{N}-\sqrt{n-1})$ with high probability. If we assume
further the entries of $X$ have subgaussian decay, then the smallest singular
value of $TX-A$ is at least of the order $\sqrt{N}-\sqrt{n-1}$ with high
probability, which is an essentially optimal estimate.
