Matrix/Row rank and column rank/Manipulations/Fact

Let ${}K$ be a field, and let ${}M$ denote an ${}m\times n$-matrix over ${}K$.

Then the

column rank coincides with the

row rank.

If ${}M$ is transformed with elementary row manipulations to a matrix ${}M'$ in the sense of fact, then the rank equals the number of relevant rows of ${}M'$.

Proof, Write another proof