On the equality row rank and column rank (Q1567441)

The different proofs of the equality of the row rank and column rank of any rectangular matrix \(A\) over a field are discussed. It is proved using vector space duals that this equality holds also over any division ring if we mean by row rank of \(A\) the dimension of the left vector space generated by the rows of \(A\) and by column rank of \(A\) the dimension of the right vector space generated by the columns of \(A\). This proof shows that the equality of row rank and column rank is a consequence of the associativity of matrix multiplication over arbitrary rings rather than commutative properties of the entries of \(A\).