initial commit

.gitignore

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+# LaTeX
+*.aux
+*.fdb_latexmk
+*.fls
+*.synctex.gz
+*.log
+*.out

README.md

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+# tbil-la
+
+Source files and built PDFs for running a Team-Based Inquiry Learning
+linear algebra course.

course-notes.tex

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+\documentclass{article}
+
+\usepackage{tbil-la}
+
+\usepackage[left=1in,right=1in,top=1in,bottom=1in]{geometry}
+
+
+
+
+\begin{document}
+
+\begin{readinessAssuranceOutcomes}
+\item Solve a system of linear equations (including finding a basis of the solution space if it is homogeneous) by interpreting as an augmented matrix and row reducing \standardList{E1, E2, E3, E4}.
+\item State the definition of linear independence, and determine if a set of vectors is linearly dependent or independent \standardList{V5}.
+\item State the definition of a spanning set, and determine if a set of vectors spans a vector space or subspace \standardList{V6, V7}.
+\item State the definition of a basis, and determine if a set of vectors is a basis \standardList{V8, V9}.
+\end{readinessAssuranceOutcomes}
+
+\begin{readinessAssuranceResources}
+\item TODO
+\end{readinessAssuranceResources}
+
+
+\newpage
+\begin{center}{\bf Readiness Assurance Test} \end{center}
+\begin{enumerate}[1)]
+
+\item Which of the following is a solution to the system of linear equations
+\begin{align*}
+x+3y-z &= 2\\
+2x+8y+3z &=-1 \\
+-x-y+9z &= -10
+\end{align*}
+
+\begin{multicols}{4}
+\begin{enumerate}[(a)]
+\item $\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$
+\item $\begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}$
+\item $\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$
+\item $\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}$
+\end{enumerate}
+\end{multicols}
+
+
+\item Find a basis for the solution set of the following homogeneous system of linear equations
+\begin{align*}
+x+2y+-z-w &= 0 \\
+-2x-4y+3z+5w &= 0
+\end{align*}
+\begin{multicols}{4}
+\begin{enumerate}[(a)]
+\item $\left\{ \begin{bmatrix} 2 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 0 \\ 3 \\ 1 \end{bmatrix} \right\}$
+\item $\left\{ \begin{bmatrix} 2 \\ 2 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 3 \\ 0 \end{bmatrix} \right\}$
+\item $\left\{ \begin{bmatrix} 2 \\ 1 \\ 3 \\ 1 \end{bmatrix} \right\}$
+\item None of these are a basis.
+\end{enumerate}
+\end{multicols}
+
+
+\item Determine which property applies to the set of vectors $$\left\{ \begin{bmatrix}  1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right\} \subset \IR^3.$$
+\begin{enumerate}[(a)]
+\item It does not span and is linearly dependent
+\item It does not span and is linearly independent
+\item It spans but it is linearly dependent
+\item It is a basis of $\IR^3$.
+\end{enumerate}
+
+\item Determine which property applies to the set of vectors $$\left\{ \begin{bmatrix}  1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} \right\}\subset \IR^3.$$
+\begin{enumerate}[(a)]
+\item It does not span and is linearly dependent
+\item It does not span and is linearly independent
+\item It spans but it is linearly dependent
+\item It is a basis of $\IR^3$.
+\end{enumerate}
+
+\item Determine which property applies to the set of vectors $$\left\{ \begin{bmatrix}  1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 0 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 3 \\ 3 \\ -3 \end{bmatrix}\right\}\subset \IR^3.$$
+\begin{enumerate}[(a)]
+\item It does not span and is linearly dependent
+\item It does not span and is linearly independent
+\item It spans but it is linearly dependent
+\item It is a basis of $\IR^3$.
+\end{enumerate}
+
+\item Determine which property applies to the set of vectors $$\left\{ \begin{bmatrix}  2 \\ 2 \\ -1 \end{bmatrix}, \begin{bmatrix} -3 \\ 1 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 5 \\ -4 \end{bmatrix}\right\}\subset \IR^3.$$
+\begin{enumerate}[(a)]
+\item It does not span and is linearly dependent
+\item It does not span and is linearly independent
+\item It spans but it is linearly dependent
+\item It is a basis of $\IR^3$.
+\end{enumerate}
+
+\item Find a basis for the subspace of $\IR^4$ spanned by the vectors ...
+
+\item Suppose you know that every vector in $\IR^5$ can be written as a linear combination of the vectors $\{\vec{v}_1, \ldots, \vec{v}_n\}$.  What can you conclude about $n$?
+\begin{enumerate}[(a)]
+\item $n \leq 5$
+\item $n=5$
+\item $n \geq 5$
+\item $n$ could be any positive integer
+\end{enumerate}
+
+\item Suppose you know that every vector in $\IR^5$ can be written uniquely as a linear combination of the vectors $\{\vec{v}_1, \ldots, \vec{v}_n\}$.  What can you conclude about $n$?
+\begin{enumerate}[(a)]
+\item $n \leq 5$
+\item $n=5$
+\item $n \geq 5$
+\item $n$ could be any positive integer
+\end{enumerate}
+
+\item Suppose you know that every vector in $\IR^5$ can be written uniquely as a linear combination of the vectors $\{\vec{v}_1, \ldots, \vec{v}_n\}$.  What can you conclude about the set $\{\vec{v}_1, \ldots, \vec{v}_n\}$?
+\begin{enumerate}[(a)]
+\item It does not span and is linearly dependent
+\item It does not span and is linearly independent
+\item It spans but it is linearly dependent
+\item It is a basis of $\IR^3$.
+\end{enumerate}
+
+\end{enumerate}
+
+\newpage
+
+Day 1
+\begin{app}
+A {\bf linear transformation} is a map between vector spaces that preserves the vector space operations.  More precisely, if $V$ and $W$ are vector spaces, a map $T:V\rightarrow W$ is called a linear transformation if
+\begin{enumerate}
+\item $T(\vec{v}+\vec{w}) = T(\vec{v})+T(\vec{w})$ for any $\vec{v},\vec{w} \in V$
+\item $T(c\vec{v}) = cT(\vec{v})$ for any $c \in \IR$, $\vec{v} \in V$.
+\end{enumerate}
+In other words, a map is linear if one can do vector space operations before applying the map or after, and obtain the same answer.
+
+$V$ is called the {\bf domain} of $T$ and $W$ is called the {\bf co-domain} of $T$.
+
+\begin{enumerate}[1)]
+\item Determine if each of the following maps are linear transformations
+\begin{enumerate}[(a)]
+\item $T_1 : \IR^2 \rightarrow \IR$ given by $T_1\left(\begin{bmatrix} a \\ b \end{bmatrix} \right) = \sqrt{a^2+b^2}$
+\item $T_2 : \IR^3 \rightarrow \IR^2$ given by $T_2\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} x-z \\ y \end{bmatrix}$
+\item $T_3: \P_d \rightarrow \P_{d-1}$ given by $T_3(f(x)) = f^\prime(x)$.
+\item $T_4: C(\IR) \rightarrow C(\IR)$ given by $T_4(f(x)) = f(-x)$
+\item $T_5: \P \rightarrow \P$ given by $T_5(f(x)) = f(x)+x^2$
+\end{enumerate}
+
+\item Suppose $T: \IR^3 \rightarrow \IR^2$ is a linear transformation, and you know $T\left(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 2 \\ 1 \end{bmatrix} $ and $T\left(\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} -3 \\ 2 \end{bmatrix} $.  Compute each of the following:
+\begin{enumerate}[(a)]
+\item $T\left(\begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix}\right)$
+\item $T\left(\begin{bmatrix} 0 \\ 0 \\ -2 \end{bmatrix}\right)$
+\item $T\left(\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}\right)$
+\item $T\left(\begin{bmatrix} -2 \\ 0 \\ 5 \end{bmatrix}\right)$
+\end{enumerate}
+
+\item Suppose $T: \IR^4 \rightarrow \IR^3$ is a linear transformation.  What is the smallest number of vectors needed to determine $T$?  In other words, what is the smallest number $n$ such that there are $\vec{v}_1,\ldots,\vec{v}_n \in \IR^4$ and given  $T(\vec{v}_1), \ldots, T(\vec{v}_n)$ you can determine $T(\vec{w})$ for {\bf any} $\vec{w} \in \IR^2$?
+\end{enumerate}
+
+Fix an ordered basis for $V$.  Since every vector can be written {\em uniquely} as a linear combination of basis vectors, a linear transformation $T:V \rightarrow W$ corresponds exactly to a choice of where each basis vector goes.  For convenience, we can thus encode a linear transformation as a matrix, with one column for the image of each basis vector (in order).
+
+\begin{enumerate}[1)]
+\item Let $T: \IR^3 \rightarrow \IR^2$ be a linear transformation with
+\begin{align*}
+T\left(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right) &= \begin{bmatrix} 3 \\ 2\end{bmatrix} &
+T\left(\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right) &= \begin{bmatrix} -1 \\ 4\end{bmatrix} &
+T\left(\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right) &= \begin{bmatrix} 5 \\ 0\end{bmatrix}
+\end{align*}
+Write the matrix corresponding to this linear transformation with respect to the standard ordered basis.
+
+\item Let $T: \IR^3 \rightarrow \IR^2$ be a linear transformation with
+\begin{align*}
+T\left(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right) &= \begin{bmatrix} 3 \\ 2\end{bmatrix} &
+T\left(\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right) &= \begin{bmatrix} -1 \\ 4\end{bmatrix} &
+T\left(\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right) &= \begin{bmatrix} 5 \\ 0\end{bmatrix}
+\end{align*}
+Write the matrix corresponding to this linear transformation with respect to the ordered basis $\left\{ \begin{bmatrix} 2 & 1 & 1 \end{bmatrix} , \begin{bmatrix} -1 & -1 & 3 \end{bmatrix} , \begin{bmatrix} 0 & 1 & 2 \end{bmatrix} \right\}$
+
+\item Let $D: \P_3 \rightarrow \P_2$ be the derivative map (recall this is a linear transformation).  Write the matrix corresponding to $D$ with respect to the ordered basis $\{1,x,x^2,x^3\}$.
+
+\end{enumerate}
+\end{app}
+
+Day 2
+\begin{app}
+Let $T: V \rightarrow W$ be a linear transformation.
+\begin{itemize}
+\item $T$ is called {\em injective} or {\em one-to-one} if $T$ does not map two distinct values to the same place.  More precisely, $T$ is injective if $T(\vec{v}) \neq T(\vec{w})$ whenever $\vec{v} \neq \vec{w}$.
+\item $T$ is called {\em surjective} or {\em onto} if every element of $W$ is mapped to by an element of $V$.  More precisely, for every $\vec{w} \in W$, there is some $v \in V$ with $T(\vec{v})=\vec{w}$.
+\end{itemize}
+
+\begin{enumerate}[1)]
+\item Let $T: \IR^3 \rightarrow \IR^2$ be given by the matrix $\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}$.  Determine if $T$ is injective, surjective, both, or neither.
+\item Let $T: \IR^2 \rightarrow \IR^3$ be given by the matrix $\begin{bmatrix} 1 & 0 &0  \\ 0 & 1 & 0 \end{bmatrix}$.  Determine if $T$ is injective, surjective, both, or neither.
+\end{enumerate}
+
+
+We also have two important sets called the {\em kernel} of $T$ and the {\em image} of $T$.
+\begin{align*}
+\ker T &= \left\{ \vec{v} \in V\ \big|\ T(\vec{v})=0\right\} \\
+\Im T &= \left\{ \vec{w} \in W\ \big|\ \text{there is some }v\in V \text{ with } T(\vec{v})=\vec{w}\right\}
+\end{align*}
+
+\begin{enumerate}[1)]
+\item Let $T: \IR^3 \rightarrow \IR^2$ be given by the matrix $\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}$ (for the standard basis).  Find the kernel and image of $T$.
+\item Let $T: \IR^2 \rightarrow \IR^3$ be given by the matrix $\begin{bmatrix} 1 & 0 &0  \\ 0 & 1 & 0 \end{bmatrix}$ (for the standard basis).  Find the kernel and image of $T$.
+\end{enumerate}
+
+
+\begin{enumerate}[1)]
+\item Describe surjective linear transformations in terms of the image.
+\item Describe injective linear transformations in terms of the kernel.
+\end{enumerate}
+
+Let $T: \IR^3 \rightarrow \IR^2$ be the linear transformation given by the matrix $A=\begin{bmatrix} 3 & 4 & -1 \\ 1 & 2 & 1 \end{bmatrix}$ (for the standard basis).
+\begin{enumerate}[1)]
+\item Write a system of equations whose solution set is the kernel.
+\item Compute $\RREF(A)$ and solve the system of equations.
+\item Compute the kernel of $T$
+\item Find a basis for the kernel of $T$
+\end{enumerate}
+
+Let $S: \IR^3 \rightarrow \IR^2$ be the linear transformation given by the matrix $B=\begin{bmatrix} 3 & 4 & 1 \\ 1 & 2 & 4 \\ 5 & 8 & 9 \end{bmatrix}$ (for the standard basis).
+\begin{enumerate}[1)]
+\item Write a system of equations whose solution set is the kernel.
+\item Compute $\RREF(A)$ and solve the system of equations.
+\item Compute the kernel of $T$
+\item Find a basis for the kernel of $T$
+\end{enumerate}
+
+Let $T: \IR^3 \rightarrow \IR^3$ be the linear transformation given by the matrix $A=\begin{bmatrix} 3 & 4 & -1 \\ 1 & 2 & 1 \end{bmatrix}$ (for the standard basis).
+\begin{enumerate}[1)]
+\item Find a set of vectors that span the image of $T$
+\item Find a basis for the image of $T$.
+\end{enumerate}
+
+Let $S: \IR^3 \rightarrow \IR^3$ be the linear transformation given by the matrix $B=\begin{bmatrix} 3 & 4 & 1 \\ 1 & 2 & 4 \\ 5 & 8 & 9  \end{bmatrix}$ (for the standard basis).
+\begin{enumerate}[1)]
+\item Find a set of vectors that span the image of $T$
+\item Find a basis for the image of $T$.
+\end{enumerate}
+
+\end{app}
+
+
+Day 3
+\begin{app}
+Let $T: \IR^n \rightarrow \IR^m$ be a linear map with matrix $A \in M_{m,n}$ (for the standard basis).  Consider the following statements about $T$
+\begin{enumerate}[(a)]
+\item $T$ is injective
+\item $T$ is not injective
+\item $T$ is surjective
+\item $T$ is not surjective
+\item The system of linear equations given by the augmented matrix $\begin{bmatrix}[c|c]A & \vec{b} \end{bmatrix}$ has a solution for all $\vec{b} \in \IR^m$
+\item The system of linear equations given by the augmented matrix $\begin{bmatrix}[c|c]A & \vec{b} \end{bmatrix}$ has a unique solution for all $\vec{b} \in \IR^m$
+\item The system of linear equations given by the augmented matrix $\begin{bmatrix}[c|c] A & \vec{0} \end{bmatrix}$ has a non-trivial solution.
+\item The columns of $A$ span $\IR^m$
+\item The columns of $A$ are linearly independent
+\item The columns of $A$ are a basis of $\IR^m$
+\item Every column of $\RREF(A)$ is a pivot column
+\item $\RREF(A)$ has a non-pivot column
+\item $\RREF(A)$ has $n$ pivot columns
+\end{enumerate}
+
+\begin{enumerate}[1)]
+\item Sort these statements into groups of equivalent statements.
+\item Gallery walk--switch boards with a different team.  If they have two things grouped together that you know are not equivalent, write a reason or counter-example on a sticky note.
+\item Update your team's groupings based on feedback.
+\item Repeat?
+\item Can you add any statements to any groups?
+\end{enumerate}
+\end{app}
+
+
+
+
+
+\end{document}

tbil-la.sty

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+\usepackage{enumerate,amssymb,tikz,amsmath,amsthm,multicol,hyperref}
+
+\newcommand{\IR}{\mathbb{R}}
+\renewcommand{\P}{\mathcal{P}}
+\renewcommand{\Im}{{\rm Im\ }}
+\theoremstyle{definition}
+\newtheorem{definition}{Definition}
+\newtheorem{example}{Example}
+\newtheorem{app}{Application Activity}
+\DeclareMathOperator{\RREF}{RREF}
+\makeatletter
+\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
+  \hskip -\arraycolsep
+  \let\@ifnextchar\new@ifnextchar
+  \array{#1}}
+\makeatother
+
+
+\newenvironment{readinessAssuranceOutcomes}{
+\section*{Readiness Assurance Outcomes}
+
+Before beginning this module, each student should be able to...
+\begin{itemize}
+}{
+\end{itemize}
+}
+
+\newcommand{\standardList}[1]{\textbf{(Standard(s) #1)}}
+
+\newenvironment{readinessAssuranceResources}{
+\subsection*{Readiness Assurance Resources}
+The following resources will help you prepare for this module.
+\begin{itemize}
+}{
+\end{itemize}
+}