Applied edits

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

@@ -57,18 +57,19 @@ Can you develop geometric intuition for their sum and difference?
 \pagebreak
 
 \definition{Euclidean Norm}
-In general, a \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
-Usually, one thinks of a norm as a \say{length metric} on a vector space. \\
+A \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
+Usually, one thinks of a norm as a way of mesuring \say{length} in a vector space. \\
 The norm of a vector $v$ is written $||v||$. \\
 
 \vspace{2mm}
 
-We usually use the \textit{euclidean norm} when we work in $\mathbb{R}^n$. \\
-If $v \in \mathbb{R}^n$, the euclidean norm is defined as follows:
+We usually use the \textit{Euclidean norm} when we work in $\mathbb{R}^n$. \\
+If $v \in \mathbb{R}^n$, the Euclidean norm is defined as follows: \\
+If $v = [v_1, v_2, ..., v_n]$,
 $$
 	||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}
 $$
-This is simply an application of the pythagorean theorem.
+This is simply an application of the Pythagorean theorem.
 
 \problem{}
 Compute the euclidean norm of
@@ -78,4 +79,10 @@ Compute the euclidean norm of
 \end{itemize}
 
 \vfill
+
+\problem{}
+Show that $a \cdot a$ is $||a||^2$.
+
+\vfill
+
 \pagebreak