Cleanup

Advanced/Linear Algebra 101/main.tex

@@ -61,7 +61,7 @@
 	Prove the Cauchy-Schwartz inequality:
 
 	$$
-		||\langle x, y \rangle|| = ||x||~||y||
+		||x \cdot y|| = ||x||~||y||
 	$$
 
 	\vfill

Advanced/Linear Algebra 101/parts/1 vectors.tex

@@ -31,7 +31,7 @@ Can you develop geometric intuition for their sum and difference?
 	\begin{tikzpicture}[scale=1]
 
 		\draw[->]
-			(0,0) coordinate (o) -- node[below left] {$(1, 2)$}
+			(0,0) coordinate (o) -- node[below left] {$(2, -1)$}
 			(2, -1) coordinate (a)
 		;
 

Advanced/Linear Algebra 101/parts/2 dotprod.tex

@@ -40,7 +40,7 @@ Show that the dot product is
 
 
 \problem{}
-Say you have two vectors, $a$ and $b$. Show that $\langle a, b \rangle$ = $||a||~||b||\cos(\alpha)$ \\
+Say you have two vectors, $a$ and $b$. Show that $a \cdot b$ = $||a||~||b||\cos(\alpha)$ \\
 \hint{What is $c$ in terms of $a$ and $b$?}
 \hint{The law of cosines is $a^2 + b^2 - 2ab\cos(\alpha) = c^2$}
 \hint{The length of $a$ is $||a||$}
@@ -83,7 +83,7 @@ Say you have two vectors, $a$ and $b$. Show that $\langle a, b \rangle$ = $||a||
 \vfill
 
 \problem{}
-If $a$ and $b$ are perpendicular, what must $\langle a, b \rangle$ be? Is the converse true?
+If $a$ and $b$ are perpendicular, what must $a \cdot b$ be? Is the converse true?
 
 
 \vfill

Advanced/Linear Algebra 101/parts/3 matrices.tex

@@ -76,7 +76,7 @@ $$
 \pagebreak
 
 \definition{}
-We also multiply a matrix by a matrix:
+We can also multiply a matrix by a matrix:
 
 $$
 AB =
@@ -257,11 +257,11 @@ $
 \vfill
 \pagebreak
 
-The \say{transpose} operator is often used to write column vectors compactly. \\
+The \say{transpose} operator is often used to write column vectors in a compact way. \\
 Vertical arrays don't look good in horizontal text.
 
 \problem{}
-Consider the vectors $a = [1, 2, 3]^T$ and $b = [40, 50, 60]^T$ \\
+Consider the vectors $a = [1, 4, 3]^T$ and $b = [9, 1, 4]^T$ \\
 \begin{itemize}
 	\item Compute the dot product $ab$.
 	\item Can you redefine the dot product using matrix multiplication?