This paper investigates the nonasymptotic properties of the spectral norm of some random matrices with independent columns. In particular, we consider an $m\times n$ random matrix $BA$, where $A$ is an $N\times n$ random matrix with independent mean-zero subexponential entries, and $B$ is an $m\times N$ deterministic matrix. We prove that the $L_{p}$ norm of the spectral norm of $BA$ is upper bounded by $(\sqrt{m}+\sqrt{n})p$. It is remarkable that this result is independent of the dimension $N$.