From 9927c8ff43203cce0bf09b43b159e138ed774257 Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Sun, 26 Mar 2023 22:09:51 -0700
Subject: [PATCH] Edits

---
 Advanced/Linear Maps/main.tex              |  8 ++++----
 Advanced/Linear Maps/parts/2 linearity.tex |  2 +-
 Advanced/Linear Maps/parts/3 matrices.tex  | 20 +++++++++++++++++---
 3 files changed, 22 insertions(+), 8 deletions(-)

diff --git a/Advanced/Linear Maps/main.tex b/Advanced/Linear Maps/main.tex
index c8b742b..dc0c404 100755
--- a/Advanced/Linear Maps/main.tex	
+++ b/Advanced/Linear Maps/main.tex	
@@ -39,12 +39,12 @@
 
 	\section{Bonus}
 
-	\problem{}
-	Is the set of all linear maps a vector space?
-	\vfill
-
 	\definition{}
 	Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space.
 	\vfill
 
+	\problem{}
+	Is the set of all linear maps a vector space?
+	\vfill
+
 \end{document}
\ No newline at end of file
diff --git a/Advanced/Linear Maps/parts/2 linearity.tex b/Advanced/Linear Maps/parts/2 linearity.tex
index 9382198..e634f24 100644
--- a/Advanced/Linear Maps/parts/2 linearity.tex	
+++ b/Advanced/Linear Maps/parts/2 linearity.tex	
@@ -4,7 +4,7 @@
 A \textit{function} or \textit{map} $f$ from a set $A$ to a set $B$ is a rule that assigns an element of $B$ to each element of $A$. We write this as $f: A \to B$.
 
 
-\definition{This one is worth remembering}
+\definition{}
 Let $V$ and $U$ be vector spaces, and let $f: V \to U$ be a map from $V$ to $U$. \\
 We say $f$ is \textit{linear} if it satisfies the following for any $v \in V$, $u \in U$, $a \in \mathbb{R}$:
 \begin{itemize}
diff --git a/Advanced/Linear Maps/parts/3 matrices.tex b/Advanced/Linear Maps/parts/3 matrices.tex
index 104ce56..fe62669 100644
--- a/Advanced/Linear Maps/parts/3 matrices.tex	
+++ b/Advanced/Linear Maps/parts/3 matrices.tex	
@@ -16,7 +16,7 @@ Draw a $3 \times 2$ matrix.
 
 \vfill
 
-\definition{Matrices as Transformations}
+\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.}
 $$
@@ -57,7 +57,7 @@ $$
 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{}
-Compute the following \say{product}:
+Compute the following:
 
 $$
 \begin{bmatrix}
@@ -186,13 +186,27 @@ Before you start, answer the following questions:
 \problem{}<proofback>
 Show that any linear transformation can be written as a matrix.
 
+\vfill
+\pagebreak
+
+\problem{}
+Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space.
+\vfill
+
+
+\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?}
+
+
 \vfill
 \pagebreak
 
 
 \problem{}
 Does \ref{thebigtheorem} hold in arbitrary vector spaces? \\
-Repeat \ref{prooffwd} and \ref{proofback} using only axioms.
+Repeat \ref{prooffwd} and \ref{proofback} using only axioms, without assuming that we're working in $\mathbb{R}^n$.
 
 \vfill
 \pagebreak
\ No newline at end of file