z

notes.tex

@@ -70,14 +70,16 @@ Let $M$ be a binary matroid with binary representation $B\in \F_2^{n\times m}$.
 \end{lemma}
 \begin{proof}
 Let $M'$ be $M\left(\begin{bmatrix}B\\ \sigma\end{bmatrix}\right)$ and let the ground set be $E$.
-$\lambda(M')\leq \lambda(M)$ (row space). 
-Let $F^*\in \argmin_{F\subset E} \frac{|E-F|}{r'(E)-r'(F)}$.
+We have $\lambda(M')\leq \lambda(M)$ since the row space becomes larger. 
+
+Let $F^*\in \argmin_{F\subset E} \frac{|E-F|}{r'(E)-r'(F)}$. Let $r$ be the rank of $M$ and let $r'$ be the rank of $M'$.
+We can assume that $r(E)-r(F^*)\geq 1$ since otherwise we have $r'(E)-r'(F^*)\leq 1$ which implies the gap of $M'$ is 1.
 
 \begin{equation*}
 \begin{aligned}
 \sigma(M')  &=      \frac{|E-F^*|}{r'(E)-r'(F^*)}\\
             &\geq   \frac{|E-F^*|}{r(E)+1-r(F^*)}\\
-			&\geq   \frac{|E-F^*|}{2(r(E)-r(F^*)}\\
+			&\geq   \frac{|E-F^*|}{2(r(E)-r(F^*))}\\
 			&\geq   \frac{\sigma(M)}{2}
 \end{aligned}
 \end{equation*}