\documentclass{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{enumerate}
\usepackage{ifthen}
\usepackage{ulem} % for strikethrough \sout{...} command
\usepackage{color}

\setlength{\unitlength}{0.1in}

\setlength{\oddsidemargin}{-0.25in}
\setlength{\evensidemargin}{-0.25in}
\setlength{\topmargin}{-0.25in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\topskip}{0.0in}
\setlength{\textheight}{9.75in}
\setlength{\textwidth}{7in}

\newcommand{\bdot}{\, {\scriptstyle{\bullet}} \,}
\newcommand{\B}[1]{{\bf {#1}}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newcommand{\myspace}{\parbox[c][0.25in]{0.25in}{\hfill}}
\newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}}
\newcommand{\force}{{\noindent $\mbox{\;}$}}

\newcommand{\diagquotient}[2]{\displaystyle{{#1 \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop #2}}}

\begin{document}

\pagestyle{empty}

\noindent
\parbox{2.5in}{\bf Math 5230}
\hfill {\Large \bf Almost Final Exam} \hfill
\parbox{2.5in}{\bf \hfill Due by Monday at 5pm}\\
\force \hfill \hfill {\bf (Dec. $\bf 5^\mathrm{th}$)} \qquad \  \   \force

\force \hfill {\bf\color{red}Extended Edition} \hfill \force

\vspace{0.2in}

\noindent {\bf DIRECTIONS:} You may use books, notes, software, and existing online resources to complete this exam. You may {\bf not} talk to anyone (except me) about these problems or ask for help online. 

\vspace{0.1in}

\noindent {\sc Notation:} Recall that $P_n = \{a_nx^n+\cdots +a_2x^2+a_1x+a_0 \;|\; a_0,\dots,a_n \in \mathbb{R}\}$ is the space of all polynomials of degree at most $n$, $\mathrm{Hom}(V,W) = \mathcal{L}(V,W)$ is the space of all linear transformations from $V$ to $W$, and $\mathbb{R}[x]$ is the space of all polynomials with real coefficients.

\vspace{0.2in}

\noindent {\bf\large 1.} Let $A = \begin{bmatrix} 2 & 1 & 2 & 4 & 6 \\ 0 & 0 & 2 & 0 & 2 \\ 2 & 1 & 4 & 6 & 8 \end{bmatrix}$.
\begin{enumerate}[(a)]
\item Find an invertible matrix $P$ such that $PA=R$ is the reduced row echelon form of $A$.
\item Write $P$ as a product of elementary matrices.
\end{enumerate}

\vspace{0.05in}

\noindent {\bf\large 2.} Let $S_1 = \{ x^3-1, x^2+2x, 2x^3-x^2-2x-2 \}$. Define $W=\{ ax^3+bx^2+cx+d \;|\; a+2b-c+d=0 \}$. 
\begin{enumerate}[(a)]
\item Show $W$ is a subspace.
\item Find a basis for $W$.
\item Show $\mathrm{span}(S_1) \subseteq W$. Are they equal? Why or why not?
\end{enumerate}

\vspace{0.05in}

\noindent {\bf\large 3.} Let $T:P_2 \to \mathbb{R}^{2 \times 3}$ be defined by $T(ax^2+bx+c) = \begin{bmatrix} a & b & c \\ b & c & a \end{bmatrix}$. Let $\beta_1 = \{1,x,x^2\}$ (note the order) and $\beta_2 = \{1-x,1+x,x-x^2 \}$. Also, let $\gamma_1 = \{E_{11},E_{12},E_{13},E_{21},E_{22},E_{23}\}$ and 
$$\gamma_2 = \left\{ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \right\}.$$
\begin{enumerate}[(a)]
\item Prove $T$ is linear.
\item Prove that $\beta_2$ is a basis for $P_2$ and $\gamma_2$ is a basis for $\mathbb{R}^{2 \times 3}$.
\item Find $[T]_{\beta_1}^{\gamma_1}$.
\item Find a basis for the kernel and range of $T$. State $T$'s rank and nullity. Is $T$ one-to-one, onto, both or neither?
\item Find the change of coordinate matrix from $\beta_1$ to $\beta_2$.
\item Find $[T]_{\beta_2}^{\gamma_2}$.
\item Find $[x^2+2x+1]_{\beta_2}$. Then compute $[T(x^2+2x+1)]_{\gamma_2}$ using part \sout{(e)} (f)  [Actually, (e) is useful too.].
\end{enumerate}

\vspace{0.05in}

\noindent {\bf\large 4.} Let $A = \begin{bmatrix} 1 & 3 & 3 \\ 3 & 1 & 3 \\ 3 & 3 & 1 \end{bmatrix}$. 
\begin{enumerate}[(a)]
\item Looking at $A$, we can immediately conclude it is diagonalizable over the real numbers. Why? 
\item Find the characteristic polynomial of $A$ and $A$'s eigenvalues as well as their algebraic and geometric multiplicities. [Note: You probably will want to have technology factor your polynomial for you.]
\item Find an {\bf orthogonal} matrix $P$ that diagonalizes $A$.
\end{enumerate}

\vspace{0.05in}

\noindent {\bf\large 5.} We run an experiment and find the following data points: $(x,y,z)=$ $(1,2,3)$, $(0,-1,2)$, $(2,1,1)$, and $(-1,1,0)$. Let $f(x,y)=A\sin(xy)+B\cos(x+y)+C$. Find constants $A,B,C$ such that $z=f(x,y)$ best fits the given data (give a least squares solution).
[Note: Doing this by hand would be a very very bad idea. You can give an approximate solution (i.e. use decimal approximations of $A,B,C$).]

\newpage

\noindent {\bf\large 6.} Let $T,U:V \to W$ be a linear transformations between vector spaces $V$ and $W$ (vector spaces over a field $\mathbb{F}$).
\begin{enumerate}[(a)]
\item Let $S \subseteq W$ and suppose $S$ is linearly independent. Show that $T^{-1}(S) = \{ \B v \in V \;|\; T(\B v) \in S \}$ is linearly independent.
\item Recall that the annihilator of a subset $X \subseteq V$ is $A(X) = \{ f \in V^* \;|\; f(\B x)=0 \mbox{ for all } \B x \in X\}$ and $T^t:W^* \to V^*$ is defined by $T^t(f)= f \circ T$. Show that $A(T(V))=\mathrm{Ker}(T^t)$ (i.e. the annihilator of the range is equal to the kernel of the transpose map). 
\item Suppose that neither $T$ nor $U$ are the zero transformation. Also, assume that $T(V) \cap U(V) = \{\B 0\}$ (i.e. the ranges of $T$ and $U$) have a trivial intersection. Prove that $\{T,U\}$ is linearly independent as a subset of $\mathcal{L}(V,W)$.
\color{red}
\item Suppose that $V$ and $W$ are finite dimensional and use a previous part to show that $T$ is an isomorphism if and only if $T^t$ is an isomorphism.
\item Let $U,T : V \to V$ be commuting linear operators (i.e. $U \circ T = T \circ U$). Show that any eigenspace of $U$ is $T$-invariant.
\end{enumerate}

\vspace{0.05in}

\noindent {\large \bf 7.} Let $T$ be a linear operator defined on a finite dimensional inner product space. Show $T^*(V)\subseteq \mathrm{Ker}(T)^\perp$ (the range of the adjoint is contained in the orthogonal complement of the kernel of the operator). {\color{red} Bonus: Show equality.}

\vspace{0.1in}

\noindent {\large \bf 8.} 
                      If the statement is always true, write ``True'' and {\bf prove it}. 
                      If the statement is never true, write ``False'' and {\bf prove} that it can never be true.
                      If the statement is true in some cases and false in others, write ``Possible'' then give an 
                      {\it example} {\bf and} a {\it counter-example}.
                      
                      \vspace{0.1in}
                      
                      You may skip 4.
\begin{enumerate}[(a)]

\item Let $A \in \mathbb{R}^{5 \times 5}$ be a matrix with characteristic polynomial $f(t) = t(t-1)(t-2)(t-3)(t-4)$.\\
         {\bf TRUE / POSSIBLE / FALSE}: $A$ is diagonalizable. 

\item Let $A \in \mathbb{F}^{n \times n}$ where $\mathbb{F}$ is field and $n$ is a positive integer.\\
         {\bf TRUE / POSSIBLE / FALSE}: $A$ can be written as a finite product of elementary matrices.  
         
\item Let $W = \{ax^2+bx+c \;|; a+b=0, c=1 \}$\\
         {\bf TRUE / POSSIBLE / FALSE}: $W$ is a subspace of $\mathbb{R}[x]$. 

\item Let $A \in \mathbb{C}^{n \times n}$ (for some positive integer $n>1$) and suppose $\det(A)=3$. Let $E$ be the $n \times n$ elementary matrix associated with the row operation: add 5 times row 2 to row 1.\\
         {\bf TRUE / POSSIBLE / FALSE}:  $\det(-E^TA^{-2}) = -\dfrac{1}{9}$. 

\item Let $A \in \mathbb{R}^{3 \times 3}$ have characteristic polynomial $f(t)=t^3-2t^2+t$. In addition suppose that the eigenspace for eigenvalue $\lambda=1$ is $E_1 = \left\{ \begin{bmatrix} a \\ b \\ -a \end{bmatrix} {\Bigg|} a,b \in \mathbb{R} \right\}$.\\
         {\bf TRUE / POSSIBLE / FALSE}: $A$ is diagonalizable. 

\item Let $n$ be a positive integer.\\
         {\bf TRUE / POSSIBLE / FALSE}: $\mathcal{L}(\mathbb{R}^3,\mathbb{R}^{n+1}) \cong \mathbb{R}^{n \times n} \oplus P_{2n}$.

\item Let $T:P_5 \to P_5$ be a one-to-one linear transformation.\\
          {\bf TRUE / POSSIBLE / FALSE}: $T$ is onto. 

\item Suppose that $A \in \mathbb{R}^{n \times n}$ and that $A{\bf x} = {\bf b}$ is inconsistent for some ${\bf b} \in \mathbb{R}^{n \times 1}$\\
         {\bf TRUE / POSSIBLE / FALSE}: $A{\bf x}={\bf 0}$ has infinitely many solutions. 

\item Let $T:\mathbb{R}^{2 \times 2} \to P_5$ be a linear transformation. Assume that $\mathrm{rank}(T)=2$.\\
         {\bf TRUE / POSSIBLE / FALSE}: $\mathrm{Ker}(T) \cong T(\mathbb{R}^{2 \times 2})$ (i.e. the kernel and range of $T$ are isomorphic).  
         
\item Let $A \in \mathbb{Z}^{3 \times 3}$ (i.e. $A$ is a $3 \times 3$ matrix with integer entries).\\
         {\bf TRUE / POSSIBLE / FALSE}: $\det(A^{-1})=\sqrt{5}$.  

\item Let $T:P_2 \to \mathbb{R}^{2 \times 3}$ be a linear transformation.\\
         {\bf TRUE / POSSIBLE / FALSE}: $\mathrm{rank}(T)=4$.  

\item Let $S \subset V$ where $V$ is an inner product space (over some field $\mathbb{F}$). Suppose that $S$ is an orthogonal set.\\
         {\bf TRUE / POSSIBLE / FALSE}: $S$ is linearly independent. 

\item Let $A \in \mathbb{C}^{n \times n}$ be a Hermitian matrix.\\
         {\bf TRUE / POSSIBLE / FALSE}: $\det(A) \in \mathbb{R}$. 

\vspace{0.05in}

\color{red}

\item Let $A \in \mathbb{R}^{3 \times 3}$ have characteristic polynomial $f(t)=t^3-2t^2-t$\\
         {\bf TRUE / POSSIBLE / FALSE}: $A$ is invertible. % false

\item Let $T:\mathbb{R}[x] \to \mathbb{R}[x]$ be a one-to-one linear transformation.\\
          {\bf TRUE / POSSIBLE / FALSE}: $T$ is onto. % possible

\item Let $T:P_3 \to \mathbb{R}^5$ be a linear transformation.\\
         {\bf TRUE / POSSIBLE / FALSE}: $T$ is one-to-one.  % possible

\item Let $T:P_3 \to \mathbb{R}^5$ be a linear transformation.\\
         {\bf TRUE / POSSIBLE / FALSE}: $T$ is onto. % false

\item Recall that a square matrix $P$ is called a permutation matrix if the rows of $P$ are just a permutation of the rows of the identity matrix. Let $P$ be a permutation matrix.\\
         {\bf TRUE / POSSIBLE / FALSE}: $P$ is orthogonal. % true

\item Let $Q \in \mathbb{R}^{n \times n}$ be an orthogonal matrix.\\
         {\bf TRUE / POSSIBLE / FALSE}: $-Q$ is orthogonal. % true


\end{enumerate}


\end{document}