Assume $f_1, f_2,...,f_n: \mathbb{R}^d\mapsto\mathbb{R}^d$ are linearly independent functions. Now let $w_1,w_2,..,w_k\in\mathbb{R}^d$ be i.i.d. Gaussian random vectors distributed as $\mathcal{N}(0,\mathbf{I}_d)$ and $v_1,\ldots,v_k\in\mathbb{R}$ are i.i.d. Radamacher random variables ($\pm 1$ with equal probability). I want to show that as long as $kd\ge n$ then with high probability the matrix

\begin{align*} \begin{bmatrix} v_1f_1(\mathbf{w}_1) & v_1f_2(\mathbf{w}_1) & \ldots & v_1f_n(\mathbf{w}_1)\\ v_2f_1(\mathbf{w}_2) & v_2f_2(\mathbf{w}_2) & \ldots & v_2f_n(\mathbf{w}_2)\\ \vdots & \vdots & \ddots & \vdots\\ v_kf_1(\mathbf{w}_k) & v_kf_2(\mathbf{w}_k) & \ldots & v_kf_n(\mathbf{w}_k) \end{bmatrix}\in\mathbb{R}^{kd\times n} \end{align*} is full rank