week 6 sections

Author: annadavismath

VSP-0020/main.tex

@@ -153,7 +153,7 @@ \subsection*{Subspaces of $\RR^n$}
 Let $S$ be any set of vectors in $\RR^n$.  Then $\mbox{span}(S)$ is a subspace of $\RR^n$.
 \end{theorem}
 
-In particular, if we take $S$ to be the single vector $\vec{v}$, we have that $\text{span}(\vec{v})$ is a subspace of $\RR^n$.  Geometrically, this subspace is a line in the direction of the vector $\vec{v}$.  Similarly, the span of two vectors is a subspace of $\RR^n$.  If the two vectors are linearly independent, then the subspace is a plane in $\RR^n$. 
+In particular, if we take $S$ to be the single vector $\vec{v}$, we have that $\text{span}(\vec{v})$ is a subspace of $\RR^n$.  Geometrically, this subspace is a line with a direction vector $\vec{v}$.  Similarly, the span of two vectors is a subspace of $\RR^n$.  If the two vectors are linearly independent, then the subspace is a plane in $\RR^n$. 
 
 Not every line or plane in $\RR^n$ is a subspace, however.  The following important result provides us with a quick way to determine that some subsets are not subspaces.
 
@@ -174,10 +174,7 @@ \subsection*{Subspaces of $\RR^n$}
 
 \section*{Practice Problems}
 
-\emph{Problems \ref{prob:Y^+1}-\ref{prob:Y^+2}}
-
-Let $Y^+$ be the set of all vectors in $\mathbb{R}^2$ whose $y$ components are non-negative.    
- \begin{problem}\label{prob:Y^+1}
+ \begin{problem}\label{prob:Y^+1} Let $Y^+$ be the set of all vectors in $\mathbb{R}^2$ whose $y$ components are non-negative.  
  Is $Y^+$ closed under vector addition?
 
  \begin{multipleChoice}
@@ -186,7 +183,7 @@ \section*{Practice Problems}
  \end{multipleChoice}
  \end{problem}
 
- \begin{problem}\label{prob:Y^+2}
+ \begin{problem}\label{prob:Y^+2} Let $Y^+$ be the set of all vectors in $\mathbb{R}^2$ whose $y$ components are non-negative.  
  Is $Y^+$ closed under scalar multiplication?
 
   \begin{multipleChoice}
@@ -197,11 +194,7 @@ \section*{Practice Problems}
 
 
 
-\emph{Problems \ref{pr:R^3axes1}-\ref{pr:R^3axes2}}
-
- Let $X$ be the set of all vectors in $\RR^3$ that lie on either the $x$-axis, the $y$-axis, or the $z$-axis.    
- 
- 	\begin{problem}\label{pr:R^3axes1}
+ 	\begin{problem}\label{pr:R^3axes1}  Let $X$ be the set of all vectors in $\RR^3$ that lie on either the $x$-axis, the $y$-axis, or the $z$-axis. 
   Is $X$ closed under vector addition?
 
   \begin{multipleChoice}
@@ -210,21 +203,16 @@ \section*{Practice Problems}
  \end{multipleChoice}
 
  	\end{problem}
- 	\begin{problem}\label{pr:R^3axes2}
+ 	\begin{problem}\label{pr:R^3axes2}  Let $X$ be the set of all vectors in $\RR^3$ that lie on either the $x$-axis, the $y$-axis, or the $z$-axis. 
  Is $X$ closed under scalar multiplication?
 
  \begin{multipleChoice}
  \choice[correct]{Yes}
  \choice{No}
  \end{multipleChoice}
  	\end{problem}
-
-
-\emph{Problems \ref{prob:closedmultchoice1}-\ref{prob:closedmultchoice2}}
-
-For each figure below, determine whether the set $V$ of vectors shown in the figure is closed under vector addition and scalar multiplication.  Justify your responses.
 
-\begin{problem}\label{prob:closedmultchoice1} 
+\begin{problem}\label{prob:closedmultchoice1} Determine whether the set $V$ of vectors shown in the figure is closed under vector addition and scalar multiplication.  Justify your responses.
 $V$ consists of all vectors in $\mathbb{R}^3$ in a slanted half-plane which has the $x$-axis as a boundary.
 
 \tdplotsetmaincoords{70}{130}
@@ -255,7 +243,7 @@ \section*{Practice Problems}
  \end{multipleChoice}
 \end{problem}
 
-\begin{problem} \label{prob:closedmultchoice2}
+\begin{problem} \label{prob:closedmultchoice2} Determine whether the set $V$ of vectors shown in the figure is closed under vector addition and scalar multiplication.  Justify your responses.
 $V$ consists of all vectors along the line, as shown.
 
 \begin{tikzpicture}[scale=0.5]
@@ -286,11 +274,11 @@ \section*{Practice Problems}
 
 
 \begin{problem}\label{prob:opposite_in_subspace}
-Prove Theorem \ref{th:opposite_in_subspace}.
+Prove that if $V$ is a subspace of $\RR^n$, then for any vector $\vec{v} \in V$, the opposite vector, $-\vec{v}$, is also in $V$. (Theorem \ref{th:opposite_in_subspace})
 \end{problem}
 
 \begin{problem}\label{prob:null(A)_is_subspace}
-Let $A$ be an $m \times n$ matrix.  Let $V$ be the subset of $\RR^n$ consisting of all vectors $\vec{x}$ such that $A \vec{x} = \vec{0}$.  Prove that $V$ is a subspace of $\RR^n$.  (Note that this subspace is called the null space of the matrix $A$ and we will denote it $\mbox{null}(A)$.)
+Let $A$ be an $m \times n$ matrix.  Let $V$ be the subset of $\RR^n$ consisting of all vectors $\vec{x}$ such that $A \vec{x} = \vec{0}$.  Prove that $V$ is a subspace of $\RR^n$.  (This subspace is called the null space of the matrix $A$.  We will denote it $\mbox{null}(A)$.)
 \end{problem}
 
 \end{document}

VSP-0030/main.tex

@@ -275,12 +275,10 @@ \subsection*{Definition of a Basis}
 
 
 \section*{Practice Problems}
-\emph{Problems \ref{prob:coordvect1}-\ref{prob:coordvect2}}
 
-Let $\mathcal{B}=\left\{\begin{bmatrix}1\\1\end{bmatrix},\begin{bmatrix}-1\\2\end{bmatrix}\right\}$ be a basis for $\RR^2$.  (Do a mental verification that $\mathcal{B}$ is a basis.)  For each $\vec{v}$ given below, find the coordinate vector for $\vec{v}$ with respect to $\mathcal{B}$.
 
   \begin{problem}\label{prob:coordvect1}
- Vector $\vec{v}$. 
+Let $\mathcal{B}=\left\{\begin{bmatrix}1\\1\end{bmatrix},\begin{bmatrix}-1\\2\end{bmatrix}\right\}$ be a basis for $\RR^2$.  Find the coordinate vector for $\vec{v}$ with respect to $\mathcal{B}$.  
   \begin{center}
 \begin{tikzpicture}[scale=0.6]
 \draw[thin,gray!40] (-4,-2) grid (4,4);
@@ -295,7 +293,7 @@ \section*{Practice Problems}
   \end{problem}
 
   \begin{problem}\label{prob:coordvect2}
-  Vector $\vec{v}$.
+Let $\mathcal{B}=\left\{\begin{bmatrix}1\\1\end{bmatrix},\begin{bmatrix}-1\\2\end{bmatrix}\right\}$ be a basis for $\RR^2$.  Find the coordinate vector for $\vec{v}$ with respect to $\mathcal{B}$.
   \begin{center}
 \begin{tikzpicture}[scale=0.6]
 \draw[thin,gray!40] (-1,-1) grid (4,8);

VSP-0035/main.tex

@@ -126,43 +126,45 @@ \subsection*{Every Subspace of $\RR^n$ has a Basis}
 \end{proof}
 
 \section*{Practice Problems}
-\emph{Problems \ref{prob:finddimension1}-\ref{prob:finddimension3}}
 
-For each given set $S$ of vectors, find $\mbox{dim}(\mbox{span}(S))$.
-\begin{hint}
-Use Theorem \ref{th:linindandrank} of VEC-0110.
-\end{hint}
+
+% \begin{hint}
+% Use Theorem \ref{th:linindandrank} of VEC-0110.
+% \end{hint}
 
 \begin{problem}\label{prob:finddimension1}
+Find $\mbox{dim}(\mbox{span}(S))$.
 $$S=\left\{\begin{bmatrix}1\\1\\0\\1\end{bmatrix}, \begin{bmatrix}0\\1\\1\\1\end{bmatrix}, \begin{bmatrix}1\\0\\1\\1\end{bmatrix}, \begin{bmatrix}1\\1\\0\\1\end{bmatrix} \right\}$$
 Answer: $\mbox{dim}(\mbox{span}(S))=\answer{3}$
 \end{problem}
 
 \begin{problem}\label{prob:finddimension2}
+Find $\mbox{dim}(\mbox{span}(S))$.
 $$S=\left\{\begin{bmatrix}3\\-2\\1\\1\end{bmatrix}, \begin{bmatrix}2\\3\\3\\-2\end{bmatrix}, \begin{bmatrix}1\\-5\\-2\\3\end{bmatrix}\right\}$$
 Answer: $\mbox{dim}(\mbox{span}(S))=\answer{2}$
 \end{problem}
 
 \begin{problem}\label{prob:finddimension3}
+Find $\mbox{dim}(\mbox{span}(S))$.
 $$S=\left\{\begin{bmatrix}1\\1\\-3\end{bmatrix}, \begin{bmatrix}-3\\2\\1\end{bmatrix}, \begin{bmatrix}5\\-2\\4\end{bmatrix}\right\}$$
 Answer: $\mbox{dim}(\mbox{span}(S))=\answer{3}$
 \end{problem}
 
 
 \begin{problem}\label{prob:atmostnlinindinrnproof}
-Prove Lemma \ref{lemma:atmostnlinindinrn}.
-\begin{hint}
-Look at the proof of Theorem \ref{th:dimwelldefined}.
-\end{hint}
+Prove that if a linearly independent subset of $\RR^n$ contains $m$ vectors, then $m\leq n$. (Lemma \ref{lemma:atmostnlinindinrn})
+% \begin{hint}
+% Look at the proof of Theorem \ref{th:dimwelldefined}.
+% \end{hint}
 \end{problem}
 
-\begin{problem}\label{prob:matrixtimesbasisvectors} 
-Let $\mathcal{B}=\{\vec{v}_1, \vec{v}_2, \vec{v}_3\}$ be a basis of $\RR^3$.  Suppose $A$ is a nonsingular $3\times 3 $ matrix.  Show that $\mathcal{C}=\{A\vec{v}_1, A\vec{v}_2, A\vec{v}_3\}$ is also a basis of $\RR^3$. 
-\begin{hint}
-To show that $\mathcal{C}$ spans $\RR^3$, express $A^{-1}\vec{v}$ as a linear combination of $\vec{v}_1$, $\vec{v}_2$ and $\vec{v}_3$.
-\end{hint}
+% \begin{problem}\label{prob:matrixtimesbasisvectors} 
+% Let $\mathcal{B}=\{\vec{v}_1, \vec{v}_2, \vec{v}_3\}$ be a basis of $\RR^3$.  Suppose $A$ is a nonsingular $3\times 3 $ matrix.  Show that $\mathcal{C}=\{A\vec{v}_1, A\vec{v}_2, A\vec{v}_3\}$ is also a basis of $\RR^3$. 
+% \begin{hint}
+% To show that $\mathcal{C}$ spans $\RR^3$, express $A^{-1}\vec{v}$ as a linear combination of $\vec{v}_1$, $\vec{v}_2$ and $\vec{v}_3$.
+% \end{hint}
 
-\end{problem}
+% \end{problem}
 
 
 

week6.tex

@@ -0,0 +1,16 @@
+\documentclass{xourse}
+
+\input{preamble.tex}
+
+ \title{Week 6}
+
+\begin{document}
+\begin{abstract}
+\end{abstract}
+\maketitle
+
+%\part{Week 6 assignments}
+\activity{VSP-0020/main}
+\activity{VSP-0030/main}
+\activity{VSP-0035/main}
+\end{document}