Borel subgroup is a semidirect product of the subgroup of all unitriangular matrices and the maximal split torus

Question

Suppose $G = GL(n,q)$ and $B\leqslant G$ is the Borel subgroup, $N \leqslant G$ is the subgroup of all monomial matrices and $T=B\cap N$ is a maximal split torus. I am trying to understand why $B=U \rtimes T$. It is clear that $T$ is a subgroup of diagonal matrices, so $$T \cong \underbrace{C_{q-1} \times ... \times C_{q-1}}_n,$$ but don't understand, why any element of $B$ can be uniquely written as a product $ut$ for $u \in U$, $t \in T$. I understand this is stupid, but how do we show the uniqueness?