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.