Rank of product of matrices

[Riemann]

Let $A, B$ be $m \times n$ and $n \times k$ matrices respectively with entries over some field. Prove that

\[{\rm rank} (AB) \geq {\rm rank} (A) + {\rm rank}(B) -n\]

[Tolaso J Kos]

It holds that

$${\rm nul} (T_1 T_2) \leq {\rm nul} (T_1) + {\rm nul} (T_2)$$

where $T_1, \; T_2$ are the corresponding linear transformations.

Proof: The proof of the lemma is based on the rank - nullity theorem.


Based upon the above lemma we have that

\begin{align*}
{\rm rank} \left ( T_1 T_2 \right ) + n &= k - {\rm nul} \left ( T_1 T_2 \right ) +n \\
&\geq n - {\rm nul} (T_1) + k - {\rm nul} (T_2) \\
&={\rm rank} (T_1) + {\rm rank} (T_2)
\end{align*}