Minor edits

Advanced/Linear Maps/parts/1 spaces.tex

@@ -16,13 +16,12 @@ A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
 
 \textbf{Note:}
 The same symbols are used for additions and multiplications in both $\mathbb{F}$ and $V$. \\
-Be careful, since \textit{these are different operations!} \\
+\textit{These are different operations}, so be aware of the context of each $+$ and $\times$.
-Make sure you're aware of the context of each $+$ and $\times$ as you work through this handout.
 
 \vspace{5mm}
 
 Vector addition and multiplication must have the following properties. \\
-Note that $x, y, z \in V$ and $a, b\in \mathbb{F}$.
+In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$.
 
 \vspace{2mm}
 
@@ -52,13 +51,19 @@ Note that $x, y, z \in V$ and $a, b\in \mathbb{F}$.
 		Closure			& \multicolumn{2}{c}{$ax \in V$} \\
 		Distributivity	& $a(x+y)~$&$~ax+ay$	\\
 						& $(a+b)x~$&$~ax+bx$	\\
-		Compatibility$^*$	& $(ab)x~$&$~x(ba)$		\\
+		Compatibility$^*$	& $(ab)x~$&$~a(bx)$		\\
 		Identity		& $a+0~$&$~a$
 	\end{tabular}
 \end{center}
 \end{minipage}
 
-\vspace{5mm}
+\vspace{2mm}
+
+$^*$ Remember that $a, b \in \mathbb{F}$ and $x \in V$. Thus, $(ab)$ is scalar multiplication and $(bx)$ is vector multiplication. \\
+Compatibility is \textit{not} associativity, it is \say{compatibility of vector and scalar multiplication.}
+
+\vfill
+\pagebreak
 
 \definition{}
 There is a good chance you are familiar with basic vector arithmetic. \\