The question: Let A be an m*n matrix whose kernel is 0; that is, the only solution of the equation Ax=0 is x=0. What is the rank of A?
what did I think: Ax is injective transformation, so the rank of A can't be m.
The answer is Rank(A)=n, but I don't know how to achieve it. Can somebody help me?
The Rank-nullity theorem states that $\operatorname {rk}A+\operatorname {nullity}A=n$. Since the kernel is trivial, $\operatorname {nullity}A=0$.
(Incidentally, this means $m\ge n$.)