The Truncated, Upper-Triangle Pascal Matrix, Linear Independence and Results in Matroid Theory

Let q be a power of a prime p. Matrices over Fq in which every subset of basis size of the columns are independent, are of interest in coding theory, matroid theory, and projective geometry. For any positive integer m ≤ p and bijection σ : N≤q−1 ∪ {0} → Fq, we show that the m× (q + 1) matrix Hq,m, with {Uq}i,j = 