How to find a set of $2k-1$ vectors such that each element of set is an element of $\mathcal{R}$ and any $k$ elements of set are linearly independent?

Question

Let's define set $A=\{\vec{a_1},\vec{a_2},...,\vec{a_n} \}$ where $\vec{a_i} \in R^k$. How can I find a set $A$ such that any $k$ elements of the set are linearly independant?

Answer 1

In $\Bbb R^k$ consider $v_t=\pmatrix{1&t&t^2&t^3&\cdots&t^{k-1}}$ for $t\in\Bbb R$. Any $k$ distinct $v_t$ are linearly independent. (Consider Vandermonde determinants.)