Before you read (Section 2 - 6)

book/linearalgebra.tex

@@ -1001,6 +1001,24 @@ \section*{Vector Combinations}
 			objects (like line segments or polygons).
 	\end{itemize}
 
+	Before you read this section, make sure you are comfortable with the following tasks.
+	Please do the ``Quick Check'' problem to see if you are comfortable with each
+	task.
+	\begin{itemize}
+		\item Distinguishing between two kinds of quantifiers
+
+		\item[] Quick Check: Let $\mathbb{L}$ be the set of English letters. Let $S=\Set
+			{e,\ell,a}$. Write down a subset $T_{1}$ of $\mathbb{L}$ so that \emph{some}
+			elements of $S$ are in $T_{1}$. Write down another subset $T_{2}$ of
+			$\mathbb{L}$ so that \emph{all} elements of $S$ are in $T_{2}$.
+
+		\item Visualizing vector addition and scalar multiplication.
+
+		\item[] Quick Check: On a coordinate sheet, draw the vector $\vec v_{1} = (1,
+			0)$ and $\vec v_{2} = (1,1)$. Draw the set of all scalar multiples of
+			$\vec v_{1}$. Draw the vector $\vec v_{1} + \vec v_{2}$.
+	\end{itemize}
+
 	\input{modules/module2.tex}
 	\input{modules/module2-exercises.tex}
 
@@ -1594,6 +1612,27 @@ \section*{Lines and Planes}
 		\item How to find linearly independent subsets.
 	\end{itemize}
 
+	Before you read this section, make sure you are comfortable with the following tasks.
+	Please do the ``Quick Check'' problem to see if you are comfortable with each
+	task.
+	\begin{itemize}
+		\item Understanding the definition of linear combination
+
+		\item[] Quick Check: Write down the set of all linear combinations of
+			$\mat{2 \\ 3}$ and $\mat{4 \\ 6}$ using set-builder notation. Is $\mat{2 \\ 2}$a
+			linear combination of these two vectors?
+
+		\item Solving simple systems of linear equations
+
+		\item[] Quick Check: Solve the system of linear equations
+			$\systeme{x+2y=7, -x+3y=8}$.
+
+		\item Working with lines and planes
+
+		\item[] Quick Check: Let $\mathcal{L}$ be the line passing through $(1,0)$ and $(2,4)$.
+			Write down the vector form of $\mathcal{L}$.
+	\end{itemize}
+
 	\input{modules/module3.tex}
 	\input{modules/module3-exercises.tex}
 \end{module}
@@ -2376,6 +2415,27 @@ \section*{Linear Independence and Dependence, Creating Examples}
 		\item The \emph{normal form} of lines, planes, and hyperplanes.
 	\end{itemize}
 
+	Before you read this section, make sure you are comfortable with the following tasks.
+	Please do the ``Quick Check'' problem to see if you are comfortable with each
+	task.
+	\begin{itemize}
+		\item Working with triangle ratios
+
+		\item[] Quick Check: Define $\cos$, $\sin$, and $\tan$ of an arbitrary angle
+			(not necessarily acute).
+
+		\item Using Pythagoras' theorem
+
+		\item[] Quick Check: In triangle $\mathbf{ABC}$, $\widehat{ABC}=90^{\circ}$.
+			Suppose the length of $\overline{AB}$ and $\overline{BC}$ are $5$ and
+			$12$ respectively. What's the length of $\overline{AC}$?
+
+		\item Working with spans and translated spans
+
+		\item[] Quick Check: Express the line passing through $(2,3)$ and $(4,6)$ as a span or a translated
+			span.
+	\end{itemize}
+
 	\input{modules/module4.tex}
 	\input{modules/module4-exercises.tex}
 \end{module}
@@ -3040,6 +3100,24 @@ \section*{Dot Product}
 		\item How to project a vector onto a line.
 	\end{itemize}
 
+	Before you read this section, make sure you are comfortable with the following tasks.
+	Please do the “Quick Check” problem to see if you are comfortable with each task.
+	\begin{itemize}
+		\item Working with dot product
+
+		\item[] Quick Check: Let $\vec x=(4,2)$, $y=(2,-4)$. Calculate
+			$\vec x\cdots \vec y$. What does this tell you about $\vec x$ and $\vec y$?
+
+		\item Calculating the norm of a vector
+
+		\item[] Let $\vec x=(7,-24)$. Calculate $\abs{\vec x}$.
+
+		\item Working with normal form of a line
+
+		\item[] Let $\mathcal{L}$ be given by $\mat{3 \\ 2}\cdot (\vec x-\mat{4 \\ 7}
+			)=0$. Write down a normal vector and a direction vector of $\mathcal{L}$.
+	\end{itemize}
+
 	\input{modules/module5.tex}
 	\input{modules/module5-exercises.tex}
 
@@ -3479,6 +3557,35 @@ \section*{Projections}
 		\item How to find a basis for and the dimension of a subspace.
 	\end{itemize}
 
+	Before you read this section, make sure you are comfortable with the following tasks.
+	Please do the “Quick Check” problem to see if you are comfortable with each task.
+	\begin{itemize}
+		\item Working with spans
+
+		\item[] Quick Check: Let $S=\Span\Set{\mat{4 \\ 2}, \mat{8 \\ 4}}$. What's the
+			geometric meaning of $S$? If $\vec v_{1},\vec v_{2}\in S$, is $\vec v_{1}
+			+\vec v_{2}\in S$?
+
+		\item Working with simple definitions
+
+		\item[] Quick Check: A subset $\mathbb{S}$ of $\mathbb{Z}$ is called an $I$-subset
+			if for all $s\in\mathbb{S}$ and $r\in\mathbb{Z}$, we have $rs\in\mathbb{S}$.
+			Is the set of all even numbers an $I$-subset? Is the set of all odd numbers
+			an $I$-subset? Justify your answer.
+
+		\item Verifying if a set is linearly (in)dependent
+
+		\item[] Quick Check: Let $\vec v_{1}=\mat{1 \\ 0 \\ 0}$, $\vec v_{2}=\mat{4 \\ 2 \\ 0}$,
+			and $\vec v_{3} = \mat{0 \\ 4 \\ 2}$. Is the set $\Set{\vec v_1, \vec v_2, \vec v_3}$
+			linearly independent?
+
+		\item Verifying if a set spans the whole space
+
+		\item[] Quick Check: Let $\vec v_{1},\vec v_{2}, \vec v_{3}$ be given as in
+			the previous Quick Check problem. Prove that $\Span\Set{\vec v_1,\vec v_2,\vec v_3}
+			=\R^{3}$.
+	\end{itemize}
+
 	\input{modules/module6.tex}
 	\input{modules/module6-exercises.tex}
 \end{module}

book/modules/module14-exercises.tex

@@ -1,87 +1,151 @@
 \begin{exercises}
 	\begin{problist}
-		\prob Let $\mathcal T:\R^{2}\to\R^{2}$ be defined by
-		$\mathcal T\mat{x\\y}=\matc{3x-y\\x-\tfrac{1}{4}y}$. Find the volume
-		of $\mathcal T(C_{2})$.
+		\prob Let $\mathcal{T}:\R^{2}\to\R^{2}$ be defined by $\mathcal{T}\mat{x\\y}
+		=\matc{3x-y\\x-\tfrac{1}{4}y}$. Find the volume of $\mathcal{T}(C_{2})$.
 
-		\prob Let $\mathcal S:\R^{3}\to\R^{3}$ be defined by
-		$\mathcal S\mat{x\\y\\z}=\matc{2x+y+z\\x-\tfrac{1}{2}y\\z}$. Find
-		the volume of $\mathcal S(C_{3})$.
+		\prob Let $\mathcal{S}:\R^{3}\to\R^{3}$ be defined by
+		$\mathcal{S}\mat{x\\y\\z}=\matc{2x+y+z\\x-\tfrac{1}{2}y\\z}$. Find the volume
+		of $\mathcal{S}(C_{3})$.
 
-		\prob Let $\mathcal T:\R^{2}\to\R^{2}$ be defined by
-		$\mathcal T\mat{x\\y}=\matc{x+2y\\-x-y}$.
+		\prob Let $\mathcal{T}:\R^{2}\to\R^{2}$ be defined by $\mathcal{T}\mat{x\\y}
+		=\matc{x+2y\\-x-y}$.
 		\begin{enumerate}
-			\item draw $\mathcal{E}$ and $\mathcal{T}(\mathcal{E})$ and
-				then determine whether $\mathcal{T}$ is orientation
-				preserving or orientation reversing.
+			\item Draw $\mathcal{E}$ and $\mathcal{T}(\mathcal{E})$ and then determine
+				whether $\mathcal{T}$ is orientation preserving or orientation reversing.
 
-			\item Find $\det(\mathcal T)$.
+			\item Find $\det(\mathcal{T})$.
 		\end{enumerate}
 
-		\prob For each linear transformation defined below, find its
-		determinant.
+		\prob For each linear transformation defined below, find its determinant.
 		\begin{enumerate}
-			\item $\mathcal S:\R^{2}\to\R^{2}$, where $\mathcal S$
-				shortens
+			\item $\mathcal{S}:\R^{2}\to\R^{2}$, where $\mathcal{S}$ shortens
 				every vector by a factor of $\tfrac{2}{3}$.
 
-			\item $\mathcal R:\R^{2}\to\R^{2}$, where $\mathcal R$
-				is rotation counter-clockwise by $90^{\circ}$.
+			\item $\mathcal{R}:\R^{2}\to\R^{2}$, where $\mathcal{R}$ is rotation
+				counter-clockwise by $90^{\circ}$.
 
-			\item $\mathcal F:\R^{2}\to\R^{2}$, where $\mathcal F$
-				is reflection across the line $y=-x$.
+			\item $\mathcal{F}:\R^{2}\to\R^{2}$, where $\mathcal{F}$ is
+				reflection across the line $y=-x$.
 
-			\item $\mathcal G:\R^{2}\to\R^{2}$, where
-				$\mathcal G(\vec x)=\mathcal{P}(\vec x)+
-				\mathcal{Q}(\vec x)$ and where $\mathcal{P}$ is projection onto the
-				line $y=x$ and $\mathcal{Q}$ is projection onto
-				the line $y=-\tfrac{1}{2}x$.
+			\item $\mathcal{G}:\R^{2}\to\R^{2}$, where
+				$\mathcal{G}(\vec x)=\mathcal{P}(\vec x)+ \mathcal{Q}(\vec x)$
+				and where $\mathcal{P}$ is projection onto the line $y=x$ and
+				$\mathcal{Q}$ is projection onto the line $y=-\tfrac{1}{2}x$.
 
-			\item $\mathcal T:\R^{3}\to\R^{3}$, where
-				$\mathcal T\mat{x\\y\\z}=\matc{x-y+z\\z+x-\tfrac{1}{3}y\\z}$.
+			\item $\mathcal{T}:\R^{3}\to\R^{3}$, where
+				$\mathcal{T}\mat{x\\y\\z}=\matc{x-y+z\\z+x-\tfrac{1}{3}y\\z}$.
 
-			\item $\mathcal J:\R^{3}\to\R^{3}$, where
-				$\mathcal J\mat{x\\y\\z}=\matc{0\\0\\x+y+z}$.
+			\item $\mathcal{J}:\R^{3}\to\R^{3}$, where
+				$\mathcal{J}\mat{x\\y\\z}=\matc{0\\0\\x+y+z}$.
 
-			\item $\mathcal K\circ \mathcal H:R^{2}\to\R^{2}$, where
-				$\mathcal H\mat{x\\y}=\matc{x+2y\\-x-y}$,
+			\item $\mathcal{K}\circ \mathcal{H}:\R^{2}\to\R^{2}$, where
+				$\mathcal{H}\mat{x\\y}=\matc{x+2y\\-x-y}$,
 
-				and $\mathcal K\mat{x\\y}=\matc{-x-2y\\x+y}$.
+				and $\mathcal{K}\mat{x\\y}=\matc{-x-2y\\x+y}$.
 		\end{enumerate}
 
 		\prob Let $A=\mat{2&3\\1&5}$.
 		\begin{enumerate}
 			\item Use elementary matrices to find $\det(A)$.
 
-			\item Draw a picture of the parallelogram given by the rows
-				of $A$. 
+			\item Draw a picture of the parallelogram given by the rows of $A$.
 				\label{PROBMOD14-rows}
 
-			\item Draw a picture of the parallelogram given by the columns
-				of $A$.
+			\item Draw a picture of the parallelogram given by the columns of $A$.
 				\label{PROBMOD14-cols}
-			\item How do the areas of the parallelograms drawn in parts \ref{PROBMOD14-rows} and
-				\ref{PROBMOD14-cols} relate?
+
+			\item How do the areas of the parallelograms drawn in parts \ref{PROBMOD14-rows}
+				and \ref{PROBMOD14-cols} relate?
 		\end{enumerate}
 
 		\prob Let $A=\mat{1&2&0\\0&2&1\\1&2&3}$.
 		\begin{enumerate}
-			\item \label{Module14-q8} Use elementary matrices to
-				find $\det(A)$.
+			\item \label{Module14-q8} Use elementary matrices to find $\det(A)$.
 
 			\item Find $\det(A^{-1})$.
 
 			\item Find $\det(A^{T})$, and compare your answer with
 				\ref{Module14-q8}. Are they the same? Explain.
 		\end{enumerate}
 
-		\prob Let $A$ be an $n \times n$ matrix that can be decomposed into
-		the product of elementary matrices.
+		\prob Let $A$ be an $n \times n$ matrix that can be decomposed into the
+		product of elementary matrices.
 		\begin{enumerate}
 			\item What is $\Rank(A)$? Justify your answer.
 
 			\item What is $\Null(A^{-1})$? Justify your answer.
 		\end{enumerate}
-	\end{problist}
-\end{exercises}
+		\prob Anna and Ella are studying the relationship between determinant and
+		volume. In particular, they are studying $\mathcal{S}:\mathbb{R}^{3}\rightarrow
+		\mathbb{R}^{3}$ defined by $\mathcal{S}\mat{x \\ y \\ z}=\mat{4x \\ 2z \\ 0}$,
+		and $\mathcal{T}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$ defined by
+		$\mathcal{T}\mat{x \\ y \\ z}=\mat{2x \\ 8z}$.
+
+		For each conversation below, (a) evaluate Anna and Ella's arguments as \emph{correct},
+		\emph{mostly correct}, or \emph{incorrect}; (b) point out where each argument
+		makes correct/incorrect statements; (c) give a correct numerical value
+		for the determinant or explain why it doesn't exist.
+		\begin{enumerate}
+			\item \emph{Anna says:}
+
+				Since the image of $C_{3}$ under $\mathcal{S}$ is the
+				parallelepiped generated by
+				$\mat{4 \\ 0 \\ 0},\mat{0 \\ 0 \\ 0}, and \mat{0 \\ 2 \\ 0}$, which
+				is 2-dimensional parallelogram, the volume of
+				$\mathcal{S}(C_{3})$ is just the area of this parallelogram, which
+				is 8. Thus, $\det(\mathcal{S})=8$.
+
+				\emph{Ella says:}
 
+				$\det(\mathcal{S})$ is undefined, because $\mathcal{S}$ is not
+				invertible.
+
+			\item \emph{Anna says:}
+
+				Since the image of $C_{3}$ under $\mathcal{T}$ is the
+				parallelepiped generated by $\mat{2 \\ 0}$, $\mat{0 \\ 0}$, and
+				$\mat{0 \\ 8}$, which is a parallelogram in $\mathbb{R}^{2}$,
+				the signed volume of $\mathcal{T}(C_{3})$ is just the signed
+				area of this parallelogram, which is 16. Thus,
+				$\det(\mathcal{T})=16$.
+
+				\emph{Ella says:}
+
+				$\det(\mathcal{T})$ is undefined, because $\det(\mathcal{T})$ is
+				only defined when the domain and codomain of $\mathcal{T}$ are
+				the same.
+		\end{enumerate}
+		\begin{solution}
+			\begin{enumerate}
+				\item \emph{Anna's argument is incorrect.}
+
+					\emph{Reason:} Since $\mathcal{S}$ is a linear
+					transformation on $\R^{3}$, its determinant is given by the signed
+					change of \emph{3-dimensional volume}. Anna's argument is incorrect
+					because she considered the 2-dimensional volume of $\mathcal{S}
+					(C_{3})$.
+
+					\emph{Ella's argument is incorrect.}
+
+					\emph{Reason:} The determinant is defined for all linear
+					transformations from $\R^{n}$ to $\R^{n}$, no matter whether
+					it is invertible or not.
+
+					Finally, $\det(\mathcal{S})=0$, because since $\mathcal{S}(C_{3}
+					)$ is a \emph{2-dimensional} object in $\R^{3}$, its \emph{3-dimensional
+					volume} is $0$. Therefore, $\VolChange(\mathcal{S})=0$, and we
+					conclude that $\det(\mathcal{S})=0$.
+
+				\item \emph{Anna's argument is incorrect.}
+
+					\emph{Reason:} The determinant function is only defined for
+					linear transformations with same domain and codomain.
+
+					\emph{Ella's argument is correct.}
+
+					Finally, $\det(\mathcal{T})$ is undefined, because the domain
+					and codomain of $\mathcal{T}$ are not the same.
+			\end{enumerate}
+		\end{solution}
+	\end{problist}
+\end{exercises}