Say $f_1, ..., f_n$ are linearly independent functions of the form $f_i:\mathcal{X} \rightarrow \mathbb{R}$ where the cardinality of $\mathcal{X}$ is larger or equal to $n$. Can we always find $x_1, ..., x_n \in \mathcal{X}$ such that the matrix $(f_i(x_j))_{i,j} \in \mathbb{R}^{n \times n}$ is invertible?
My guess is that this should be possible, but I cannot prove it. My intuition comes from the special case where $\mathcal{X}$ is finite, i.e. $\mathcal{X} := \{x_1, ..., x_N\}$. We can form a matrix $(f_i(x_j))_{i,j} \in \mathbb{R}^{n \times N}$ that has linearly independent rows because the functions $f_i$ are linearly independent. We can then use the fact that the row and column ranks are always equal.
