Edited Linear Algebra 101

Advanced/Linear Algebra 101/main.tex

@@ -107,37 +107,42 @@
 	\end{tikzpicture}
 	\end{center}
 
+	\vfill
 
+	\problem{}
+	If $a$ and $b$ are perpendicular, what must $\langle a, b \rangle$ be? Is the converse true?
 
 
 	\vfill
 	\pagebreak
 
-	\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
+	\section{Bonus}
-	\begin{center}
-	\begin{tikzpicture}[scale=1]
-		\draw[dashed,->] (-0.5,0) -- (4,0) node[right]{};
-		\draw[dashed,->] (0,-0.5) -- (0,3) node[above]{};
 
-		\draw[->] (0,0) -- (1,2) node[right]{};
+	\problem{}
-		\draw[->] (0,0) -- (3,0.5) node[above]{};
+	Show that the euclidean norm satisfies the triangle inequalty:
-	\end{tikzpicture}
+	$$
-	\end{center}
+	||x+y|| \leq ||x|| + ||y||
-	\end{minipage}
+	$$:
-	\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
-	\begin{center}
-	\begin{tikzpicture}[scale=1]
-		\draw[dashed,->] (-0.5,0) -- (4,0) node[right]{};
-		\draw[dashed,->] (0,-0.5) -- (0,3) node[above]{};
 
-		\draw[->] (0,0) -- (3,1) node[right]{};
+	\vfill
-		\draw[->] (0,0) -- (3,0.5) node[above]{};
-	\end{tikzpicture}
-	\end{center}
-	\end{minipage}
 
+	\problem{}
+	Show that the eucidean norm satisfies the reverse triangle inequality:
 
+	$$
+		||x - y|| \geq |~||x|| - ||y||~|
+	$$
 
+	\vfill
+
+	\problem{}
+	Prove the Cauchy-Schwartz inequality:
+
+	$$
+		||\langle x, y \rangle|| = ||x||~||y||
+	$$
+
+	\vfill
 
 
 \end{document}