Week 10

Author: annadavismath

DET-0010/main.tex

@@ -16,9 +16,9 @@ \section*{Finding the Determinant}
 
 In this section we will define a function that assigns to each square matrix $A$ a scalar output called the \dfn{determinant of $A$}.  We will denote the determinant of $A$ by $\det{A}$.  For a matrix with real number entries, the output of the determinant function will always be a real number.
 
-One important property of the determinant is its connection to matrix inverses.  We will find that a matrix $A$ is singular if and only if $\det{A}=0$.  For nonsingular matrices, we will establish a formula that gives the inverse of a matrix exclusively in terms of determinants.  This property will be addressed in detail in \href{https://ximera.osu.edu/linearalgebradzv3/LinearAlgebraInteractiveIntro/DET-0040/main}{Properties of the Determinant}, and \href{https://ximera.osu.edu/linearalgebradzv3/LinearAlgebraInteractiveIntro/DET-0060/main}{Determinants and Inverses of Nonsingular Matrices}.
+One important property of the determinant is its connection to matrix inverses.  We will find that a matrix $A$ is singular if and only if $\det{A}=0$.  %For nonsingular matrices, we will establish a formula that gives the inverse of a matrix exclusively in terms of determinants.  This property will be addressed in detail in \href{https://ximera.osu.edu/linearalgebradzv3/LinearAlgebraInteractiveIntro/DET-0040/main}{Properties of the Determinant}, and \href{https://ximera.osu.edu/linearalgebradzv3/LinearAlgebraInteractiveIntro/DET-0060/main}{Determinants and Inverses of Nonsingular Matrices}.
 
-Geometrically speaking, the determinant of a matrix of a linear transformation is the factor by which the area (or volume or hypervolume) is scaled by the transformation.  This will be discussed in \href{https://ximera.osu.edu/linearalgebradzv3/LinearAlgebraInteractiveIntro/DET-0070/main}{Determinants as Areas and Volumes}. 
+Geometrically speaking, the determinant of a matrix of a linear transformation is the factor by which the area (or volume or hypervolume) is scaled by the transformation.  %This will be discussed in \href{https://ximera.osu.edu/linearalgebradzv3/LinearAlgebraInteractiveIntro/DET-0070/main}{Determinants as Areas and Volumes}. 
 
 \subsection*{Cofactor Expansion Along the Top Row}
 To start from the beginning, let us define the determinant of a $1\times 1$ matrix.
@@ -47,15 +47,19 @@ \subsection*{Cofactor Expansion Along the Top Row}
 \begin{example}\label{ex:threebythreedet1}
 Find $\det{A}$ if 
 $$A=\begin{bmatrix}3&-2&1\\5&-1&2\\1&4&1\end{bmatrix}$$
+
 \begin{explanation}
+We will use three color-coded copies of the matrix $A$
+$$A=\begin{bmatrix}{\color{blue}3}&-2&1\\5&{\color{blue}-1}&{\color{blue}2}\\1&{\color{blue}4}&{\color{blue}1}\end{bmatrix},\quad A=\begin{bmatrix}3&{\color{red}-2}&1\\{\color{red}5}&-1&{\color{red}2}\\{\color{red}1}&4&{\color{red}1}\end{bmatrix},\quad A=\begin{bmatrix}3&-2&{\color{brown}1}\\{\color{brown}5}&{\color{brown}-1}&2\\{\color{brown}1}&{\color{brown}4}&1\end{bmatrix}$$
+
 \begin{align*}
-\det{A}&=(3)\begin{vmatrix}-1&2\\4&1\end{vmatrix}-(-2)\begin{vmatrix}5&2\\1&1\end{vmatrix}+(1)\begin{vmatrix}5&-1\\1&4\end{vmatrix}\\
+\det{A}&={\color{blue}(3)\begin{vmatrix}-1&2\\4&1\end{vmatrix}}-{\color{red}(-2)\begin{vmatrix}5&2\\1&1\end{vmatrix}}+{\color{brown}(1)\begin{vmatrix}5&-1\\1&4\end{vmatrix}}\\
 &=(3)(-1-8)-(-2)(5-2)+(1)(20+1)\\
 &=-27+6+21\\
 &=0
 \end{align*}
 
-\youtube{aR2w-viFcvI}
+%\youtube{aR2w-viFcvI}
 \end{explanation}
 \end{example}
 
@@ -125,7 +129,7 @@ \subsection*{Cofactor Expansion Along the Top Row}
 &=57
 \end{align*}
 
-\youtube{YIJq7TqncyU}
+%\youtube{YIJq7TqncyU}
 \end{explanation}
 \end{example}
 
@@ -300,66 +304,57 @@ \subsection*{Determinants of Some Special Matrices}
 
 \section*{Practice Problems}
 
-\emph{Problems \ref{prob:2x2det1}-\ref{prob:laplace}}
-
-Find the determinant of each matrix.
-
-  \begin{problem}\label{prob:2x2det1}
+  \begin{problem}\label{prob:2x2det1} Find the determinant.
   $$A=\begin{bmatrix}4&-2\\3&7\end{bmatrix}$$
   Answer:
   $$\det{A}=\answer{34}$$
   \end{problem}
 
-  \begin{problem}\label{prob:2x2det2}
+  \begin{problem}\label{prob:2x2det2} Find the determinant.
   $$B=\begin{bmatrix}5&-1&0\\0&3&-2\\1&-1&2\end{bmatrix}$$
   Answer:
   $$\text{det}(B)=\answer{22}$$
   \end{problem}
 
-  \begin{problem}\label{prob:laplace}
+  \begin{problem}\label{prob:laplace} Find the determinant.
   $$C=\begin{bmatrix}1&-2&0&0&0\\0&-4&1&1&0\\3&0&-1&0&1\\0&0&4&1&0\\-1&-2&0&0&0\end{bmatrix}$$
    Answer:
   $$\det(C)=\answer{12}$$
  \end{problem}
 
 
-\begin{problem}\label{prob:toprowexp2x2}
-Show that Definition \ref{def:toprowexpansion} is consistent with Definition \ref{def:twobytwodet} by verifying that both produce the same result when applied to a $2\times 2$ matrix.
-\end{problem}
+% \begin{problem}\label{prob:toprowexp2x2}
+% Show that Definition \ref{def:toprowexpansion} is consistent with Definition \ref{def:twobytwodet} by verifying that both produce the same result when applied to a $2\times 2$ matrix.
+% \end{problem}
 
 \begin{problem}\label{prob:detOfTrans}
-    Prove Theorem \ref{th:detoftrans}.
+    Prove that for a square matrix $A$, $\det{A^T}=\det{A}$. (Theorem \ref{th:detoftrans})
 \end{problem}
 
 \begin{problem}\label{prob:detrowswitch}
-Let $B'$ be a matrix obtained from $B$ of Problem \ref{prob:2x2det2} by switching the first and the second row of $B$.  Compute the determinant of $B'$.  What do you observe?
-\end{problem}
-
-\begin{problem}\label{prob:2x2rowswitchproof}
-Make a conjecture about what happens to the determinant of a matrix if two rows of a matrix are switched.  Prove your conjecture for a $2\times 2$ matrix.
-\end{problem}
+Let $$A=\begin{bmatrix}-2&1&3\\0&4&0\\-1&1&2\end{bmatrix}$$
 
-\begin{problem}\label{prob:scalarmultrowdet} Let $B'$ be a matrix obtained from $B$ of Problem \ref{prob:2x2det2} by multiplying the middle row by $-3$.  Compute the determinant of $B'$.  What do you observe?
-\end{problem}
-
-\begin{problem}\label{prob:rowtimesconstant2x2proof}
-Make a conjecture about what happens to the determinant of a matrix if one of the rows is multiplied by a constant.  Prove your conjecture for a $2\times 2$ matrix.
-\end{problem}
+\begin{enumerate}
+    \item $$\text{det}A=\answer{-4}$$
+    \item Let $A_1$ be a matrix obtained from $A$ by switching the first and the second row of $A$.  
+$$\text{det}A_1=\answer{4}$$
+    What do you observe?
 
-\begin{problem}\label{prob:matrixtimesconst}
-Let $B'$ be a matrix obtained from $B$ of Problem \ref{prob:2x2det2} by multiplying $B$ by $2$.  Compute the determinant of $B'$.  What do you observe?
-\end{problem}
+    \item Let $A_2$ be a matrix obtained from $A$ by multiplying the bottom row by $-3$.
 
-\begin{problem}\label{prob:matrixtimesconstant2x2proof}
-Make a conjecture about what happens to the determinant of a matrix if the matrix is multiplied by a constant.  Prove your conjecture for a $2\times 2$ matrix.
-\end{problem}
+ $$\text{det}A_2=\answer{12}$$   
+    What do you observe?
 
-\begin{problem}\label{prob:scalarmultofrow}
-Let $B'$ be a matrix obtained from $B$ of Problem \ref{prob:2x2det2} by adding twice the third row to the first.  Compute the determinant of $B'$.  What do you observe?
-\end{problem}
+    \item Let $A_3$ be a matrix obtained from $A$  by multiplying $A$ by $2$.    
+    
+    $$\text{det}A_3=\answer{-32}$$ 
+    What do you observe?  Relate your observations to what you found in the previous part.
 
-\begin{problem}\label{prob:scalarmultofrow2x2}
-Make a conjecture about what happens to the determinant of a matrix if a multiple of one row is added to another row.  Prove your conjecture for a $2\times 2$ matrix.
+    \item Let $A_4$ be a matrix obtained from $A$ by adding twice the third row to the first.  
+    $$\text{det}A_4=\answer{-4}$$ 
+    
+    What do you observe?
+\end{enumerate}
 \end{problem}
 
 \begin{problem}\label{prob:detsumsumdetquestion}

VEC-0080/main.tex

@@ -180,7 +180,7 @@ \subsection*{Properties of the Cross Product}
 \begin{proof}
 The proof is left to the reader.  (See Practice Problem \ref{prob:corssuvnegcrossvu})
 \end{proof}
-The next theorem lists two additional properties of the cross product.  Proofs of these properties are routine and are left to the reader.  (See Practice Problems \ref{prob:scalarassocofcrossprod} and \ref{prob:distofrossprod})
+The next theorem lists two additional properties of the cross product.  Proofs of these properties are routine and are left to the reader.  %(See Practice Problems \ref{prob:scalarassocofcrossprod} and \ref{prob:distofrossprod})
 
 \begin{theorem}\label{th:crossproductproperties}
 Let $\vec{u}$, $\vec{v}$ and $\vec{w}$ be vectors in $\mathbb{R}^3$, and $k$ be a scalar, then\\
@@ -237,7 +237,7 @@ \subsubsection*{Cross Product and the Angle between Vectors}
 Let $\vec{u}$ and $\vec{v}$ be vectors in $\RR^3$.  Then
 $$\norm{\vec{u}\times\vec{v}}^2=\norm{\vec{u}}^2\norm{\vec{v}}^2-(\vec{u}\dotp\vec{v})^2$$
 \end{lemma}
-\begin{proof} The proof is left to the reader.  (See Practice Problem \ref{prob:corssprodmagnitude})
+\begin{proof} The proof is left to the reader.  %(See Practice Problem \ref{prob:corssprodmagnitude})
 \end{proof}
 
 The following theorem establishes a relationship between the magnitude of the cross product, the magnitudes of the two vectors involved in the cross product and the angle between the two vectors.  It is important to note that  the identity in this theorem involves the magnitude of the cross product, not the cross product itself.
@@ -261,76 +261,79 @@ \subsubsection*{Cross Product and the Angle between Vectors}
 
 
 \section*{Practice Problems}
-\begin{problem}\label{prob:crossik}
-$\vec{i}\times\vec{k}=$
-\begin{multipleChoice}\choice{$\vec{j}$}
-  \choice[correct]{$-\vec{j}$ }
-  \choice{neither}
-  \end{multipleChoice}
-\end{problem}
+% \begin{problem}\label{prob:crossik}
+% $\vec{i}\times\vec{k}=$
+% \begin{multipleChoice}\choice{$\vec{j}$}
+%   \choice[correct]{$-\vec{j}$ }
+%   \choice{neither}
+%   \end{multipleChoice}
+% \end{problem}
 
-\begin{problem}\label{prob:crosski}
-$\vec{k}\times\vec{i}=$
-\begin{multipleChoice}\choice[correct]{$\vec{j}$}
-  \choice{$-\vec{j}$ }
-  \choice{neither}
-  \end{multipleChoice}
-\end{problem}
+% \begin{problem}\label{prob:crosski}
+% $\vec{k}\times\vec{i}=$
+% \begin{multipleChoice}\choice[correct]{$\vec{j}$}
+%   \choice{$-\vec{j}$ }
+%   \choice{neither}
+%   \end{multipleChoice}
+% \end{problem}
 
-\emph{Problems \ref{prob:crossuv1}-\ref{prob:crossuv2}} 
 
-Find the cross product $\vec{u}\times\vec{v}$, and verify that $\vec{u}\times\vec{v}$ is orthogonal to both $\vec{u}$ and $\vec{v}$.
 
-\begin{problem}\label{prob:crossuv1}
+\begin{problem}\label{prob:crossuv1} Find the cross product $\vec{u}\times\vec{v}$, and verify that $\vec{u}\times\vec{v}$ is orthogonal to both $\vec{u}$ and $\vec{v}$.
 $\vec{u}=\begin{bmatrix}2\\-1\\4\end{bmatrix}$, $\vec{v}=\begin{bmatrix}0\\3\\1\end{bmatrix}$.
 
 Answer:
 $$\vec{u}\times\vec{v}=\begin{bmatrix}\answer{-13}\\\answer{-2}\\\answer{6}\end{bmatrix}$$
 \end{problem}
 
-\begin{problem}\label{prob:crossuv2}
-$\vec{u}=\begin{bmatrix}-1\\5\\-3\end{bmatrix}$, $\vec{v}=\begin{bmatrix}2\\-1\\-4\end{bmatrix}$.
+% \begin{problem}\label{prob:crossuv2}
+% $\vec{u}=\begin{bmatrix}-1\\5\\-3\end{bmatrix}$, $\vec{v}=\begin{bmatrix}2\\-1\\-4\end{bmatrix}$.
 
-Answer:
-$$\vec{u}\times\vec{v}=\begin{bmatrix}\answer{-23}\\\answer{-10}\\\answer{-9}\end{bmatrix}$$
-\end{problem}
+% Answer:
+% $$\vec{u}\times\vec{v}=\begin{bmatrix}\answer{-23}\\\answer{-10}\\\answer{-9}\end{bmatrix}$$
+% \end{problem}
 
-\begin{problem}\label{prob:corssuvnegcrossvu}
-Prove Theorem \ref{th:corssuvnegcrossvu}.
-\end{problem}
+% \begin{problem}\label{prob:corssuvnegcrossvu}
+% Prove Theorem \ref{th:corssuvnegcrossvu}.
+% \end{problem}
 
-\begin{problem}\label{prob:scalarassocofcrossprod}
-Prove Theorem \ref{th:crossproductproperties}\ref{item:scalarassocofcrossprod} in two different ways:
-  \begin{enumerate}
-  \item By direct computation.
-  \item (Optional) By using Theorem \ref{th:elemrowopsanddet}\ref{item:rowconstantmultanddet}. 
-  \end{enumerate}
-\end{problem}
+% \begin{problem}\label{prob:scalarassocofcrossprod}
+% Prove Theorem \ref{th:crossproductproperties}\ref{item:scalarassocofcrossprod} in two different ways:
+%   \begin{enumerate}
+%   \item By direct computation.
+%   \item (Optional) By using Theorem \ref{th:elemrowopsanddet}\ref{item:rowconstantmultanddet}. 
+%   \end{enumerate}
+% \end{problem}
 
-\begin{problem}\label{prob:distofrossprod}
-Prove Theorem \ref{th:crossproductproperties}\ref{item:distofrossprod}.
+\begin{problem}
+    Let $\vec{u}$ and $\vec{v}$ be vectors in $\RR^3$, prove that
+$$\vec{u}\times\vec{v}=-(\vec{v}\times\vec{u})$$
 \end{problem}
 
+% \begin{problem}\label{prob:distofrossprod}
+% Prove Theorem \ref{th:crossproductproperties}\ref{item:distofrossprod}.
+% \end{problem}
+
 \begin{problem}\label{prob:crossproductorthtouandv}
-Prove Theorem \ref{th:crossproductorthtouandv}.
+Let $\vec{u}$ and $\vec{v}$ be vectors of $\RR^3$.  Prove that $\vec{u}\times\vec{v}$ is orthogonal to both $\vec{u}$ and $\vec{v}$. (Theorem \ref{th:crossproductorthtouandv})
 \end{problem}
 
 \begin{problem}\label{prob:crossself}
 Prove that the cross product of any vector with itself is the zero vector.
 \end{problem}
 
-\begin{problem}\label{prob:crossprodzero}
-Suppose that $\vec{u}$ is a non-zero vector.  Let $\vec{v}=k\vec{u}$ for $k\neq 0$.  Argue that $\vec{u}\times \vec{v}=\vec{0}$ in two different ways:
-\begin{enumerate}
-\item
-By using Theorem \ref{th:crossproductsin}.
-\item (Optional) By using Theorem \ref{th:detofsingularmatrix}.
-\end{enumerate}
-\end{problem}
-
-\begin{problem}\label{prob:corssprodmagnitude} Prove the following identity for vectors $\vec{u}$ and $\vec{v}$ of $\RR^3$.
-$$\norm{\vec{u}\times\vec{v}}^2=\norm{\vec{u}}^2\norm{\vec{v}}^2-(\vec{u}\dotp\vec{v})^2$$
-(Lemma \ref{lemma:crossprodmagnitude})
-\end{problem}
+% \begin{problem}\label{prob:crossprodzero}
+% Suppose that $\vec{u}$ is a non-zero vector.  Let $\vec{v}=k\vec{u}$ for $k\neq 0$.  Argue that $\vec{u}\times \vec{v}=\vec{0}$ in two different ways:
+% \begin{enumerate}
+% \item
+% By using Theorem \ref{th:crossproductsin}.
+% \item (Optional) By using Theorem \ref{th:detofsingularmatrix}.
+% \end{enumerate}
+% \end{problem}
+
+% \begin{problem}\label{prob:corssprodmagnitude} Prove the following identity for vectors $\vec{u}$ and $\vec{v}$ of $\RR^3$.
+% $$\norm{\vec{u}\times\vec{v}}^2=\norm{\vec{u}}^2\norm{\vec{v}}^2-(\vec{u}\dotp\vec{v})^2$$
+% (Lemma \ref{lemma:crossprodmagnitude})
+% \end{problem}
 
 \end{document} 

week10.tex

@@ -0,0 +1,16 @@
+\documentclass{xourse}
+
+\input{preamble.tex}
+
+ \title{Week 10}
+
+\begin{document}
+\begin{abstract}
+\end{abstract}
+\maketitle
+
+%\part{Week 10 assignments}
+\activity{DET-0010/main}
+\activity{VEC-0080/main}
+
+\end{document}