week 9

Author: annadavismath

LTR-0020/main.tex

@@ -238,31 +238,28 @@ \section*{Practice Problems}
 $$A=\begin{bmatrix}\answer{0}&\answer{-1}\\\answer{3}&\answer{-1}\\\answer{2}&\answer{3}\end{bmatrix}$$
 \end{problem}
 
-\emph{Problems \ref{prob:standardmatrix1}-\ref{prob:standardmatrix4}}. 
-
-Find the standard matrix $A$ of each linear transformation $T:\RR^2\rightarrow\RR^2$ described below.
-
   \begin{problem}\label{prob:standardmatrix1}
+  Find the standard matrix $A$ of the linear transformation $T:\RR^2\rightarrow\RR^2$ if
   $T$ doubles the $x$ component of every vector and triples the $y$ component.
 $$A=\begin{bmatrix}\answer{2}&\answer{0}\\\answer{0}&\answer{3}\end{bmatrix}$$
   \end{problem}
 
-  \begin{problem}\label{prob:standardmatrix2}
+  \begin{problem}\label{prob:standardmatrix2} Find the standard matrix $A$ of the linear transformation $T:\RR^2\rightarrow\RR^2$ if
   $T$ reverses the direction of each vector.
   $$A=\begin{bmatrix}\answer{-1}&\answer{0}\\\answer{0}&\answer{-1}\end{bmatrix}$$
   \end{problem}
 
-  \begin{problem}\label{prob:standardmatrix5}
+  \begin{problem}\label{prob:standardmatrix5} Find the standard matrix $A$ of the linear transformation $T:\RR^2\rightarrow\RR^2$ if
   $T$ doubles the length of each vector.
     $$A=\begin{bmatrix}\answer{2}&\answer{0}\\\answer{0}&\answer{2}\end{bmatrix}$$
   \end{problem}
 
-  \begin{problem}\label{prob:standardmatrix3}
+  \begin{problem}\label{prob:standardmatrix3} Find the standard matrix $A$ of the linear transformation $T:\RR^2\rightarrow\RR^2$ if
   $T$ projects each vector onto the $x$-axis. (e.g. $T\left(\begin{bmatrix}4\\5\end{bmatrix}\right)=\begin{bmatrix}4\\0\end{bmatrix}$)
     $$A=\begin{bmatrix}\answer{1}&\answer{0}\\\answer{0}&\answer{0}\end{bmatrix}$$
   \end{problem}
 
-  \begin{problem}\label{prob:standardmatrix4}
+  \begin{problem}\label{prob:standardmatrix4} Find the standard matrix $A$ of the linear transformation $T:\RR^2\rightarrow\RR^2$ if
   $T$ projects each vector onto the $y$-axis. (e.g. $T\left(\begin{bmatrix}4\\5\end{bmatrix}\right)=\begin{bmatrix}0\\5\end{bmatrix}$)
     $$A=\begin{bmatrix}\answer{0}&\answer{0}\\\answer{0}&\answer{1}\end{bmatrix}$$
  \end{problem}

LTR-0050/main.tex

@@ -283,12 +283,8 @@ \subsection*{Rank-Nullity Theorem for Linear Transformations}
 \end{proof}
 
 \section*{Practice Problems}
-\emph{Problems \ref{prob:imagerankoflintrans1}-\ref{prob:imagerankoflintrans2}}
-
-Describe the image and find the rank for each linear transformation $T:\RR^n\rightarrow \RR^m$ with standard matrix $A$ given below.
-
-  \begin{problem}\label{prob:imagerankoflintrans1}
-  $T:\RR^5\rightarrow \RR^2$, $A=\begin{bmatrix}3&2&4&7&1\\-1&-9&7&6&8\end{bmatrix}$.
+   \begin{problem}\label{prob:imagerankoflintrans1} Describe the image and find the rank of the linear transformation
+  $T:\RR^5\rightarrow \RR^2$ induced by $A=\begin{bmatrix}3&2&4&7&1\\-1&-9&7&6&8\end{bmatrix}$.
 
   \begin{multipleChoice}
   \choice[correct]{$\mbox{im}(T)=\RR^2$}
@@ -301,8 +297,8 @@ \section*{Practice Problems}
   $\mbox{rank}(T)=\answer{2}$
   \end{problem}
 
-  \begin{problem}\label{prob:imagerankoflintrans2}
-  $T:\RR^2\rightarrow\RR^3$, $A=\begin{bmatrix}1&1\\1&1\\1&1\end{bmatrix}$
+  \begin{problem}\label{prob:imagerankoflintrans2} Describe the image and find the rank of the linear transformation
+  $T:\RR^2\rightarrow\RR^3$ induced by $A=\begin{bmatrix}1&1\\1&1\\1&1\end{bmatrix}$
 
   \begin{multipleChoice}
   \choice{$\mbox{im}(T)=\RR^3$}
@@ -320,12 +316,8 @@ \section*{Practice Problems}
 \begin{problem}\label{prob:sametrans} Suppose linear transformations $T:\RR^2\rightarrow \RR^2$ and $S:\RR^2\rightarrow \RR^2$ are such that  $\mbox{im}(T)=\mbox{im}(S)=\mbox{span}\left(\begin{bmatrix}1\\-3\end{bmatrix}\right)$.  Does this mean that $T$ and $S$ are the same transformation?  Justify your claim.
 \end{problem}
 
-\emph{Problems \ref{prob:kerandnullityT1}-\ref{prob:kerandnullityT3}}
-
-Describe the kernel and find the nullity for each linear transformation $T:\RR^n\rightarrow \RR^m$ with standard matrix $A$ given below.
-
-\begin{problem}\label{prob:kerandnullityT1}
-$T:\RR^3\rightarrow \RR^2$, $A=\begin{bmatrix}2&1&0\\-1&1&-3\end{bmatrix}$.
+\begin{problem}\label{prob:kerandnullityT1} Describe the kernel and find the nullity of the linear transformation 
+$T:\RR^3\rightarrow \RR^2$ induced by $A=\begin{bmatrix}2&1&0\\-1&1&-3\end{bmatrix}$.
 
 \begin{multipleChoice}
   \choice{$\mbox{ker}(T)=\RR^3$}
@@ -338,8 +330,8 @@ \section*{Practice Problems}
 $\mbox{nullity}(T)=\answer{1}$
 \end{problem}
 
-\begin{problem}\label{prob:kerandnullityT2}
-$T:\RR^2\rightarrow \RR^2$, $A=\begin{bmatrix}2&-1\\3&0\end{bmatrix}$.
+\begin{problem}\label{prob:kerandnullityT2} Describe the kernel and find the nullity of the linear transformation 
+$T:\RR^2\rightarrow \RR^2$ induced by $A=\begin{bmatrix}2&-1\\3&0\end{bmatrix}$.
 
 \begin{multipleChoice}
   \choice{$\mbox{ker}(T)=\RR^2$}
@@ -350,8 +342,8 @@ \section*{Practice Problems}
 $\mbox{nullity}(T)=\answer{0}$
 \end{problem}
 
-\begin{problem}\label{prob:kerandnullityT3}
-$T:\RR^3\rightarrow \RR^5$, $A=\begin{bmatrix}1&2&-1\\1&2&-1\\1&2&-1\\1&2&-1\\1&2&-1\end{bmatrix}$ 
+\begin{problem}\label{prob:kerandnullityT3} Describe the kernel and find the nullity of the linear transformation 
+$T:\RR^3\rightarrow \RR^5$ induced by $A=\begin{bmatrix}1&2&-1\\1&2&-1\\1&2&-1\\1&2&-1\\1&2&-1\end{bmatrix}$ 
 
 \begin{multipleChoice}
  \choice[correct]{$\mbox{ker}(T)$ is a plane in $\RR^3$}

week9.tex

@@ -0,0 +1,16 @@
+\documentclass{xourse}
+
+\input{preamble.tex}
+
+ \title{Week 9}
+
+\begin{document}
+\begin{abstract}
+\end{abstract}
+\maketitle
+
+%\part{Week 9 assignments}
+\activity{LTR-0020/main}
+\activity{LTR-0050/main}
+\activity{LTR-0030/main}
+\end{document}