Uncountable "relatively independent" subset of finite dimensional vector spaces over an uncountable field Let $V$ be a $n$ dimensional vector space over an uncountable field ; then does there always exist an uncountable subset $S$ of $V$ such that any $n$ vectors of $S$ are linearly independent ? ( I can prove that there always exist a countably infinite subset with such property but having trouble in proving whether there is an uncountable set or not )  

The points on the curve $t\mapsto (1,t,t^2,t^3,\ldots,t^{n-1})$ satisfy this. The determinant of any matrix whose rows consist of $n$ distinct points on this curve is nonzero, as it is the Vandermonde matrix of the elements.
