A wide Gaussian matrix has full row rank almost surely

A fixed proper real or complex linear subspace $L⊊F^{\mathrm{\ell }}$ has probability zero for one Gaussian row: choose an orthogonal/unitary coordinate change sending $L$ into $\{z_1=0\}$, use [F2], and integrate the density over this coordinate subspace. It is Lebesgue-null: inside each bounded box $[-M,M{]}^d$ it lies in a coordinate hyperplane coverable by a box of width $2\epsilon$ and bounded remaining sides, whose volume tends to zero with $\epsilon$; [F4] makes that hyperplane null, and countable subadditivity covers the whole hyperplane.

Induct on the number $j\le k$ of revealed rows. The first row is nonzero almost surely by step 1.1 applied to the zero subspace. The dependence event for $j$ rows is measurable because it is the common zero set of the finitely many $j\times j$ minors. If the first $j-1$ rows are independent, their span is proper since $j-1<k<\mathrm{\ell }$; for each fixed such tuple, step 1.1 makes the section of the event that row $j$ lies in their span null. The exceptional dependent tuples are null by induction. Applying [F3] to the nonnegative indicator of dependence gives probability zero for the first $j$ rows to be dependent. The induction is finite and concludes at $j=k$.

Outside this null event, the $k$ rows are linearly independent, hence the row rank equals $k$.