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 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.

To fact