lie-algebras-and-their-representations

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -19,7 +19,7 @@ low-dimensional algebra. We begin our analysis by recalling that the elements
   f & = \begin{pmatrix} 0 & 0 \\ 1 &  0 \end{pmatrix} &
   h & = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
 \end{align*}
-form a basis of \(\mathfrak{sl}_2(K)\) and satisfy
+form a basis for \(\mathfrak{sl}_2(K)\) and satisfy
 \begin{align*}
   [e, f] & = h & [h, f] & = -2 f & [h, e] = 2 e
 \end{align*}
@@ -735,7 +735,7 @@ words\dots
 
 We call \(\lambda\) \emph{the highest weight of \(V\)}, and we call any nonzero
 \(v \in V_\lambda\) \emph{a highest weight vector}. Going back to the case of
-\(\mathfrak{sl}_2(K)\), we then constructed an explicit basis of our
+\(\mathfrak{sl}_2(K)\), we then constructed an explicit basis for our
 irreducible representations in terms of a highest weight vector, which allowed
 us to provide an explicit description of the action of \(\mathfrak{sl}_2(K)\)
 in terms of its standard basis, and finally we concluded that the eigenvalues
@@ -1121,7 +1121,7 @@ representation turns out to be quite simple.
   \end{center}
   and \(\alpha_1\) is the highest weight of \(K^3\).
 
-  On the one hand, if \(\{f_1, f_2, f_3\}\) is the dual basis of \(\{e_1, e_2,
+  On the one hand, if \(\{f_1, f_2, f_3\}\) is the dual basis for \(\{e_1, e_2,
   e_3\}\) then \(H f_i = - \alpha_i(H) \cdot f_i\) for each \(H \in
   \mathfrak{h}\), so that the weights of \((K^3)^*\) are precisely the
   opposites of the weights of \(K^3\). In other words,