Minor edits

Advanced/Linear Maps/main.tex

@@ -44,7 +44,7 @@
 	\vfill
 
 	\problem{}
-	Is the set of all linear maps a vector space?
+	Show that the set of all linear maps is a vector space.
 	\vfill
 
 \end{document}

Advanced/Linear Maps/parts/3 matrices.tex

@@ -11,14 +11,8 @@ A =
 $$
 The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix.
 
-\problem{}
-Draw a $3 \times 2$ matrix.
+We can define the product of a matrix $A$ and a vector $v$ as follows:
 
-\vfill
-
-\definition{}
-We can define the \say{product\footnotemark{}} of a matrix $A$ and a vector $v$:
-\footnotetext{This is an uncommon word to use in this context. You will soon see why.}
 $$
 Av =
 \begin{bmatrix}
@@ -34,7 +28,7 @@ Av =
 	4a + 5b + 6c
 \end{bmatrix}
 $$
-Look closely. Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
+Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
 
 $$
 Av =
@@ -56,6 +50,13 @@ $$
 
 Naturally, a vector can only be multiplied by a matrix if the number of rows in the vector equals the number of columns in the matrix.
 
+
+\problem{}
+Say you multiply a size-$m$ vector by an $m \times n$ matrix. What is the size of your result?
+
+\vfill
+
+
 \problem{}
 Compute the following:
 
@@ -156,7 +157,7 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol
 
 \vspace{2mm}
 
-Be aware that this is only a model for intuition. \\
+This is only a model for intuition, though. \\
 Make sure you understand the dot product definition on the previous page.
 
 \vspace{5mm}
@@ -167,24 +168,15 @@ Conversely, every $n \times m$ matrix represents a liner map $T: \mathbb{R}^n \t
 
 \vspace{2mm}
 
-In other words, \textbf{matrices are linear transformations}. \\
-If you only learn only one thing today, this should be it.
-
-\vfill
-
+In other words, \textbf{matrices are linear transformations}.
 \problem{}<prooffwd>
 Show that the transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\
-Before you start, answer the following questions:
-\begin{itemize}
-	\item What is $A$?
-	\item What is $v$?
-	\item What are their sizes?
-\end{itemize}
+\hint{What is $A$? What is $v$? What are their sizes?}
 
 \vfill
 
 \problem{}<proofback>
-Show that any linear transformation can be written as a matrix.
+Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$.
 
 \vfill
 \pagebreak
@@ -197,7 +189,7 @@ Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector spac
 \problem{}
 Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}p$. \\
 Find a matrix that corresponds to $D$. \\
-\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How solutions?}
+\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How so?}
 
 
 \vfill