diff git a/2rectangles/L2.tex b/2rectangles/L2.tex
index 9085e70b9f46ab657494434ac450758eda3d1150..4ffe186d2c7e7d995a6c0c3260eecd0198ea2856 100644
 a/2rectangles/L2.tex
+++ b/2rectangles/L2.tex
@@ 133,10 +133,11 @@ We will define the function $\gt(x,y)$ to be
\begin{marginfigure}
\centering
\includegraphics{figs/ge_matrix_2}
+\caption{Example matrix for $\gt(x,y)$ when $x = y = 5$ with fooling set elements of size 2 drawn.}
\end{marginfigure}
\sunote{Better to have just the figure as a PDF and use the caption for the
 text. This allows you to use the \gt command for the function name, to get a
 consistent look.}
+%\sunote{Better to have just the figure as a PDF and use the caption for the
+% text. This allows you to use the \gt command for the function name, to get a
+% consistent look.}
According to the definition of $\gt(x,y)$ the lower triangle
@@ 159,26 +160,28 @@ follows directly.
% recording ~20:20
\paragraph{Example: \disjoint}
\sunote{Didn't we define \disjoint in the last lecture notes?}

We define $x$ and $y$ to be a vector of bits
which define set membership.

\begin{align*}
 & x,y \in [0,1]^n \\
\text{and } & x_i \Leftrightarrow i \in S_x \\
 & y_i \Leftrightarrow i \in S_y
\end{align*}

the function $\disjoint(x,y)$ is defined as follows

\begin{equation*}
\disjoint(x,y) = \left\{
 \begin{array}{l l}
 1 & \quad \text{if $(x,y)$ are disjoint: } \left\{ \forall i, x_i \land y_i = 0 \right\} \\
 0 & \quad \text{otherwise}
 \end{array} \right.
\end{equation*}
+%\sunote{Didn't we define \disjoint in the last lecture notes?}
+%\ronote{In response, we did define \disjoint, but in in lecture 1
+% we did not reason about the definition with as much detail.}
+%
+%We define $x$ and $y$ to be a vector of bits
+%which define set membership.
+%
+%\begin{align*}
+% & x,y \in [0,1]^n \\
+%\text{and } & x_i \Leftrightarrow i \in S_x \\
+% & y_i \Leftrightarrow i \in S_y
+%\end{align*}
+%
+%the function $\disjoint(x,y)$ is defined as follows
+%
+%\begin{equation*}
+%\disjoint(x,y) = \left\{
+% \begin{array}{l l}
+% 1 & \quad \text{if $(x,y)$ are disjoint: } \left\{ \forall i, x_i \land y_i = 0 \right\} \\
+% 0 & \quad \text{otherwise}
+% \end{array} \right.
+%\end{equation*}
\begin{marginfigure}
@@ 191,9 +194,9 @@ $1$ & $0$ & $1$ & $1$ & $0$ & $0$ \\
$1$ & $1$ & $1$ & $0$ & $0$ & $0$ \\
\end{tabular}
\\ \vspace{1em}
Matrix for $DISJ(x,y)$ when $n=4$ showing which
+\caption{Matrix for $DISJ(x,y)$ when $n=4$ showing which
combinations will result in a $1$ or
a $0$.
+a $0$.}
\end{marginfigure}
Let us observe one of the items in the
@@ 201,7 +204,7 @@ fooling set using our example. Using the
pair $(x_1, y_1) = (01, 10)$ and $(x_2, y_2) = (10, 01)$
we can see this is part of the fooling set. This set
is interesting because $(x_1,y_1) = (\overline{x_2}, \overline{y_2})$.
In fact, the fooling st for $DISJ(x,y)$ are the bit patterns with the bits flipped;
+In fact, the fooling st for $\disjoint(x,y)$ are the bit patterns with the bits flipped;
it is the set of all $A$ and $\overline{A}$.
% recording ~33:20
@@ 348,7 +351,31 @@ rest we do not care about. We are choosing some subset of the
$r$ bits to be fixed to $1$, the rest we can pick to be anything
else. Doing this results in $2^n  2^{nr}$
\sunote{See the note that I provided in the discussion group and insert it here}
+% \sunote{See the note that I provided in the discussion group and insert it here}
+The statement that the rank of the IP matrix over GF($2$)$=2^n$ is
+not true. A proof of this can be found in Sherstov's
+notes\footnote{http://www.cs.ucla.edu/~sherstov/teaching2012winter/docs/lecture03.pdf},
+which we will briefly discuss here.
+
+The first observation is that the rank argument works over any field. So
+we can choose the field we wish to evaluate matrix rank over. In particular,
+the rank over the reals is larger than the rank over any other field.
+
+Recall that we would like to show that the inner product has deterministic
+communication complexity $n$ via the rank method. So we need to show that
+the rank of the induced matrix (where $M_{ij}=<\!\!x_i,y_j\!\!>$) is large:
+remember that IP is a function from $n$ bit string to $\{0, 1\}$.
+
+Firstly, consider the auxiliary matrix $M'=2M1$ (where $1$ is the all ons matrix).
+All this does is change $0$ to $1$ and $1$ to $1$, which is more convenient
+to work with. It is a nonsingular transformation, so $\text{rank}(M) \ge \text{rank}(M')1$.
+
+Now it is easy to see that $M'_{\text{IP}}$ on onebit strings is the Hadamard
+matrix\footnote{$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$} and the IP matrix for $n$ bit strings is merely the $n$th order
+version of this (or the $n$th tensor power of this matrix). But this matrix
+has full rank: in particular its rank is $2^n$. Therefor the rank of the IP matrix
+is at least $2^n1$.
+
%recording ~1:00:00
\section{Randomized Algorithms}
diff git a/2rectangles/figs/ge_matrix_2.ipe b/2rectangles/figs/ge_matrix_2.ipe
index f81a7475fed6030be7b07b50a0b27cba4285a8d2..1d0f5e017acde296023ff856d1336ed899dd2dbd 100644
 a/2rectangles/figs/ge_matrix_2.ipe
+++ b/2rectangles/figs/ge_matrix_2.ipe
@@ 1,7 +1,7 @@

+
\usepackage{amsmath}
@@ 233,9 +233,7 @@ h
Example matrix for $gt(x,y)$ when \\$x = y = 5$ with fooling set elements of size 2 drawn.
\\ \\
$
+$
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 \\
diff git a/2rectangles/figs/ge_matrix_2.pdf b/2rectangles/figs/ge_matrix_2.pdf
index 13407eeaaaba4a359f439f8ab4a76ce592b46cd6..0dfd92a15eab58e8118a790385cbbd4141ab87bd 100644
Binary files a/2rectangles/figs/ge_matrix_2.pdf and b/2rectangles/figs/ge_matrix_2.pdf differ