week 8 homework

Author: annadavismath

LTR-0010/main.tex

@@ -698,12 +698,31 @@ \subsection*{Linear Transformations of Subspaces of $\RR^n$}
 
 \section*{Practice Problems}
 \begin{problem}\label{prob:sum}
-
-Show that (\ref{lin2}) of Exploration \ref{init:lintransintro} holds for vectors $\begin{bmatrix}3\\4\end{bmatrix}$ and $\begin{bmatrix}-2\\1\end{bmatrix}$.
+Define $T_1$ as follows 
+$$T_1:\RR^2\rightarrow\RR^2$$
+$$T_1\left(\begin{bmatrix}
+x\\
+y
+\end{bmatrix}\right)=\begin{bmatrix}
+x-y\\
+x
+\end{bmatrix}$$
+(See Exploration \ref{init:lintransintro})
+Use vectors $\begin{bmatrix}3\\4\end{bmatrix}$ and $\begin{bmatrix}-2\\1\end{bmatrix}$ to illustrate that $T_1(\vec{u}+\vec{v}) = T_1(\vec{u})+T_1(\vec{v})$.
 \end{problem}
 
-\begin{problem}\label{prob:prob2}
-Use a counter-example to prove (\ref{t2}) of Exploration \ref{init:lintransintro}.
+\begin{problem}\label{prob:prob2} Define $T_2$ as follows
+$$T_2:\RR^2\rightarrow\RR^2$$
+$$T_2\left(\begin{bmatrix}
+x\\
+y
+\end{bmatrix}\right)=\begin{bmatrix}
+-x+y+1\\
+y-2
+\end{bmatrix}$$
+(See Exploration \ref{init:lintransintro})
+
+Use a counter-example to prove that $T_2(\vec{u}+\vec{v}) \neq T_2(\vec{u})+T_2(\vec{v})$
 \end{problem}
 
 \begin{problem}\label{prob:imageoflincomb}
@@ -714,7 +733,7 @@ \section*{Practice Problems}
 
 
 \begin{problem}\label{prob:notlinear} 
-Let $\vec{u}$ be a fixed vector.  Define $T_{\vec{u}}:\RR^2\rightarrow\RR^2$, by $T_{\vec{u}}(\vec{x})=\vec{u}-\vec{x}$.
+Let $\vec{u}$ be a fixed non-zero vector.  Define $T_{\vec{u}}:\RR^2\rightarrow\RR^2$, by $T_{\vec{u}}(\vec{x})=\vec{u}-\vec{x}$.
   \begin{enumerate}
   \item 
   Describe the effect of this transformation by sketching ${\bf x}$ and $T_{\vec{u}}({\bf x})$ for at least four vectors ${\bf x}$ and a fixed vector $\vec{u}$ of your choice.
@@ -741,22 +760,18 @@ \section*{Practice Problems}
 $$T\left(\begin{bmatrix}1\\1\\-2\end{bmatrix}\right)=\begin{bmatrix}\answer{-6}\\\answer{3}\\\answer{11}\end{bmatrix}$$
 \end{problem}
 
-\begin{problem}\label{prob:idtrans} Prove Theorem \ref{th:idlintrans}\end{problem}
-
-\begin{problem}\label{prob:zerotrans} Prove Theorem \ref{th:zerolintrans}\end{problem}
-
-\emph{Problems \ref{prob:domaincodomain1}-\ref{prob:domaincodomain2}}
+\begin{problem}\label{prob:idtrans} Prove that the identity transformation is linear. (Theorem \ref{th:idlintrans})\end{problem}
 
-For each matrix $A$ below, find the domain and the codomain of the linear transformation $T:\RR^n\rightarrow\RR^m$ induced by $A$; find and draw the image of $T$. (Hint: See Example \ref{ex:lineartrans3}.)
+\begin{problem}\label{prob:zerotrans} Prove that the zero transformation is linear. (Theorem \ref{th:zerolintrans})\end{problem}
 
-  \begin{problem}\label{prob:domaincodomain1}
+  \begin{problem}\label{prob:domaincodomain1} Find the domain and the codomain of the linear transformation $T:\RR^n\rightarrow\RR^m$ induced by $A$; find and draw the image of $T$.
   $$A=\begin{bmatrix}0&0\\1&1\\2&0\end{bmatrix}$$
   Domain: $\RR^n$, where $n=\answer{2}$.
 
   Codomain: $\RR^m$, where $m=\answer{3}$.
   \end{problem}
 
-  \begin{problem}\label{prob:domaincodomain2}
+  \begin{problem}\label{prob:domaincodomain2} Find the domain and the codomain of the linear transformation $T:\RR^n\rightarrow\RR^m$ induced by $A$; find and draw the image of $T$.
   $$A=\begin{bmatrix}3&-1\\-3&1\end{bmatrix}$$
    \end{problem}
 

LTR-0070/main.tex

@@ -789,6 +789,8 @@ \section*{Practice Problems}
     \includegraphics[height=3in]{sheep_pic.jpg}
 \end{image}
 
+$$M=\begin{bmatrix}\answer{1} & \answer{\frac{1}{\sqrt{3}}}\\\answer{0} & \answer{1}\end{bmatrix}$$
+
 %    \begin{center}
 % 		\begin{tikzpicture}[scale=2]
 % \node[inner sep=0pt, anchor=base] (gulls) at (8.75mm,0)
@@ -850,7 +852,7 @@ \section*{Practice Problems}
 \begin{problem}\label{prob:translationtrans}
 A transformation $T:\RR^2\rightarrow \RR^2$ that shifts all points in the plane horizontally or vertically by a fixed amount is called a \dfn{translation}.  Is $T$ a matrix transformation?  Prove your claim.
 \begin{hint}
-    Use Problem \ref{prb:6.4}
+   What is the image of $\vec{0}$ under $T$?
 \end{hint}
 \end{problem}
 
@@ -860,9 +862,9 @@ \section*{Practice Problems}
 \end{hint}
 \end{problem}
 
-\begin{problem} \label{prob:imageofj}
-Verify Equation (\ref{eq:imageofj}).
-\end{problem}
+% \begin{problem} \label{prob:imageofj}
+% Verify Equation (\ref{eq:imageofj}).
+% \end{problem}
 
 \begin{problem}\label{prob:fixedpoint}
 Prove that every point along the line $y=\frac{3}{5}x$ in Example \ref{ex:reflectedduck} is a fixed point.