Quadratic matrix/Rank/Invertible/Linearly independent/Fact/Proof

Proof

The equivalence of (2), (3) and (4) follows from the definition and from fact.
For the equivalence of (1) and (2), let's consider the linear mapping

defined by . The property that the column rank equals , is equivalent with the map being surjective, and this is, due to fact, equivalent with the map being bijective. Because of fact, bijectivity is equivalent with the matrix being invertible.