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

$\varphi \colon K^{n}\longrightarrow K^{n}$

defined by ${}M$. The property that the column rank equals ${}n$, 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.

To fact