prove that rankA

prove that rank(A) <= min(m, n)

A matrix AM_nm(R) is a representation of a linear transformation

f:R^mRⁿ

rank f=dim (Im f) n

rank f m

rank f min (m,n)

A matrix AM_nm(R) is a representation of a linear transformation

f:R^mRⁿ

so the image of f is a subspace of Rⁿ ,hence

rank f=dim (Im f) n

but also by the rank-nullity theorem we have

rank f m

Hence we find the desired result

rank f min (m,n)