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

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

defined by ${\displaystyle {}M}$. The property that the column rank equals ${\displaystyle {}n}$, is equivalent to the mapping being surjective, and this is, due to fact, equivalent to the mapping being bijective. Because of fact, bijectivity is equivalent to the matrix being invertible.