% f05 ribet. 9 problems (4 for math49)
% f94 lenstra. 5 problems
% s01 poonen 20 problems
% s01 vojta 13 problems
% s04 haiman 10 questions
% s96 mergman 6 problems

Ribet: You may refer to a single 2-sided sheet of notes. Your paper is
your ambassador when it is graded. Correct answers without appropriate
supporting work will be regarded skeptically. Incorrect answers
without appropriate supporting work will receive no partial credit.

\item bases for row and column space of a matrix
\item $W$ subset of $\R^{3\times 3}$ of matrices $A$ with $A^T=-A$.
  Is it a subspace, and what's it's basis?
\item Find the inverse of a matrix
\item Let $v=(1,-1,1,1)^T$ and $w=[1,1,1,1]$.  For $x\in\R^4$, let
  $T(x)=(x\cdot v)v+(x\cdot w)w$.  Show that $T$ is a linear
  transformation $\R^4\to\R^4$. Find two eigenvectors of $T$ and one
  non-zero vector $x$ such that $T(x)=0$.
\item Let $A$ be a $2\times 2$ matrix such that $A(-1,4)=4(-1,4)$
  and $A(0,1)=-4(0,1)+(-1,4)$. Find functions $x(t),y(t)$ with initial values $x(0)=-2$, $y(0)=11$ that satisfy the system of differential equations $(x',y')=A(x,y)$.
\item Suppose that $f(x)=0$ for $\pi<x<0$, $f(x)=1$ for $0\leq x\leq \pi$, and $f(x+2\pi)=f(x)$ for $x\in\R$. As usual, write the Fourier series for $f(x)$ as $a_0/2 + \sum_{m=1}^\infty(a_m\cos(mx)+b_m\sin(mx))$. Calculate the numbers $a_m$ ($m\geq 0$) and $b_m$ ($m>0$).
\item Describe all pairs of numbers $(y_0,y_0')$ such that the
  solution $y(t)$ to the initial value problem $y''-2y'-3y=0$,
  $y(0)=y_0$, $y'(0)=y_0'$ satisfies $y(t)\to 0$ as $t\to+\infty$.
\item Let $A$ be a matrix whose nullspace is $\{0\}$.  Explain carefully why each of the following statements is true: 1. The rank of $A$ equals the number of columns of $A$.  2. The rows of $A$ are linearly independent if and only if $A$ is a square matrix. 3. The product $A^TA$ is an invertible matrix.
\item Let $V$ be the vector space of continuous functions on the real
  line. Consider the inner product $f\cdot g=\int_0^1f(x)g(x)\,dx$ on
  $V$. Find a non-zero function that is orthogonal to the constant
  function $1$ and to the functions $x$ and $x^2$.
\item Solve the partial differential equation $100u_{xx}=u_t$ on the region $0<x<1$, $t>0$, subject to the boundary conditions $u(0,t)=u(1,t)$ for $t>0$ and $u(x,0)=\sin 2\pi x-\sin 5\pi x$ for $0\leq x\leq 1$.

\item Solve the system of differential equations $x_1'(t)=3x_1(t)+3x_2(t)$, $x_2'(t)=-2x_1(t)-4x_2(t)$ with initial conditions $x_1(0)=1$, $x_2(0)=3$.
\item Find three functions $y_1,y_2,y_3$ defined on $(-\infty,\infty)$ whose Wronskian is given by $W[y_1,y_2,y_3](x)=e^{4x}$. Are these functions linearly independent on $(-\infty,\infty)$?
\item Find a homogeneous third-order linear differential equation with constant coefficients that has $y(x)=3e^{-x}-\cos(2x)$ as a solution.  Explain how you found it.  What is the general solution of that differential equation?

\item Suppose $A$ is symmetric and $v_1,v_2$ two eigenvectors of $A$ corresponding to two different eigenvalues.  Show that $v_1,v_2$ are orthogonal.
\item Find the eigenvalues ofr $A=(1,1,-1;1,1,-1;-1,-1,1)$ and a basis for each of its eigenspaces. Then find three mutually orthogonal eigenvectors $v_1,v_2,v_3$.
\item Find the general real-valued solution $y(t)$ of the fourth-order differential equation $y''''+8y''+16y=0$.
\item Find the general real solution $y$ of the second-order differential equation $y''+\mu^2y=0$, where $\mu>0$. Then find that solution with the initial condition $(y(0),y'(0))=(2,3)$.
\item Find the solution $x(t)$ of the following initial value problem. $x'(t)=(3,-5;-1,-1)x(t)$, $x(0)=(6,-6)$.
\item With the inner product $\langle f,g\rangle = \int_0^{\pi/2}f(t)g(t)\,dt$, give the orthogonal projection of $f(x)=x$ onto the subspace whose orthogonal basis is $\{\sin x, \sin 3x, \sin 5x\}$.  You do not have to prove or check that this is an orthogonal basis, and you are free to use the fact that $\langle \sin nx, \sin nx\rangle = \pi/4$ for $n=1,3,5$.

Short answer:
\item Give the first row of the inverse of the matrix, if it exists.
\item The determinant of this matrix equals...
\item For what real numbers $c$ is $(1,c;0,2)$ diagonalizable.
\item Least-squares
\item True/false
  \begin{itemize}
  \item If $A$ is a square matrix, then every solution $x$ to $x'=Ax$ is a linear combination of the columns of $e^{tA}$.
  \item If $A$ is a square matrix, and the characteristic polynomial
    of $A$ is $(x-6)^2(x-7)^3$, then there exist two linearly
    independent vectors $v_1,v_2$ such that $Av_1=6v_1$ and
    $Av_2=6v_2$.
  \item If $y_1,y_2,y_3$ are solutions to the differential equation
    $y''-ty=0$ on $(-\infty,\infty)$, then the Wronskian
    $W[y_1,y_2,y_3](t)$ is the zero function.
  \item If $A$ and $B$ are symmetric $2\times 2$ matrices, then $AB$ is symmetric.
  \end{itemize}
\item Is it a subspace
  \begin{itemize}
  \item The set of solutions to $(1,2;2,4)(x,y)=(0,0)$
  \item The set of all eigenvectors of the matrix $(4,0;0,7)$,
    including the zero vector.
  \item The set of all singular $2\times 2$ matrices.
  \item The set of periodic functions $f:\R\to\R$ of period 3.
  \item The set of functions $y$ satisfying $y''+ty'+e^t y=0$ and $y(2)=0$.
  \end{itemize}
\item Which trajectory does a solution have to $x'=Ax$?
  \begin{itemize}
  \item $A=(2,1;-1,0)$
  \item $A=(-3,-1;-1,-3)$
  \item $A=(-26,-60;15,10)$
  \end{itemize}
\item Find an orthonormal basis of $\R^3$ consisting of eigenvectors of the matrix $A=(4,2,4;2,1,2;4,2,4)$ such that $(-2/3,2/3,1/3)$ is one of the vectors in the basis.
\item Find all possibilities for $x=(x_1,x_2)$ given that $x_1,x_2$ are functions satisfying $x_1'=x_1+4x_2$, $x_2'=-x_1+5x_2$, $x_1(0)=1$, $x_2(0)=1$.
\item Write the matrix $(0,5,0;1,0,2;-1,3,-1)$ as a product of elementary matrices
\item Define subspaces $V=\Span\{(1,2,3,4),(0,4,1,-1)\}$,
  $W=\Span\{(4,3,1,-6),(2,1,2,1)\}$. Find a nonzero vector in
  $V\cap W$.
\item Let $\{u,v\}$ be a basis for a vector space $W$. Under what conditions on the scalars $a,b,c,d$ is the set $\{au+bv,cu+dv\}$ also a basis of $W$?
\item Compute the determinant of the given matrix.
\item The matrix $A=(1,1,1;1,1,-1;1,-1,1)$ has eigenvalues $-1,2,2$. Find an orthogonal matrix $Q$ and a diagonal matrix $\Lambda$ such that $Q\inv A Q=\Lambda$.
\item Find the angle between $(1,2,3)$ and the $x$-axis.
\item Determine the longest interval in which the initial-value
  problem $(\tan t)y''+(t-1)y'+3y=\tan^2 t$, $y(1/2)=0$, $y'(1/2)=1$
  is certain to have a unique twice-differentiable solution. Do not
  attempt to find the solution.
\item For the equation $x'=(1,1;2,0)x$, find the fundamental matrix $\Phi(t)$ satisfying $\Phi(0)=I$.
\item For the initial value problem $x'=(0,-1;5,-2)x$, $x(0)=(1,2)$,
  describe the behaviour of the solution as $t\to\infty$. (You do not
  need to solve the system.)
\item The differential equation $x'=(3,-4,1;-1,0,-1;-2,2,0)x$ has a solution $x=(6e^{2t}+e^{-t},e^{-t},-6e^{2t})$. Find its general solution.

\item Compute an LU factorization.
\item $A$ is $3\times 3$. Given two eigenvectors and the trace,
  compute the determinant.
\item Suppose $A$ is $m\times n$, $B$ is $n\times k$, and $AB=0$. What
  can you say about the relationship between $\rank A$ and $\rank B$?
\item Set of polynomials of degree at most 5 such that
  $p(0)=p(1)=0$. Show it's a subspace, and find its dimension.
\item Compute the angle between two vectors.
\item Find an orthonormal basis for the subspace spanned by these two vectors.
\item Find all solutions of the differential equation $x^{(4)}(t)+2x^''(t)+x(t)=0$.
\item Consider the system of linear differential equations
  $x'(t)=(-1,c;1,-1)x$.
  \begin{itemize}
  \item For which values of the constant $c$ does every solution
    $x(t)=(x_1(t),x_2(t))$ have the property that
    $\lim_{t\to\infty}x_1(t)=\lim_{t\to\infty}x_2(t)=0$?
  \item For which values of $c$ does there exist a non-zero solution
    such that $x_1(t)=0$ for infinitely many values of $t$?
  \end{itemize}
\item Solve the initial value problem $x'(t)=(0,3;-2,5)x$, $x(0)=(5,4)$.
\item Find one solution of the system of linear differential equations
  $x'(t)=(2,1;1,1)x+(\sin t, 0)$.

\item Consider a matrix $A$. Find characteristic polynomial,
  eigenvalues, basis of eigenvectors, and the general solution to
  $x'=Ax$.
\item Solve a second-order system by reducing to a first-order system.
  Obtain a particular solution.
\item Give examples of: (a) two $2\times 2$ matrices $A$ and $B$ where
  $AB\neq BA$ (b) a nonzero matrix $A$ and a nonzero vector $\vec{x}$
  such that $A\vec{x}=\vec{x}$ (c) a $2\times 2$ orthogonal matrix (e)
  A vector space $V$ and a set $S$ which spans $V$ but is not a basis
  of $V$ (f) A vector $\vec{v}$ which lies in the subspace of $\R^2$
  spanned by $(1,2)$ and such that $\norm{\vec{v}}=1$.

\item True, False, or Nonsense? (Nonsense means at least one
  vocabulary word was used incorrectly.)
  \begin{itemize}
  \item If $W$ is any subspace of an inner product space, then
    $W^\perp\cap W=\{\vec{0}\}$.
  \item If $T:\mathbb{P}_2\to M^{2\times 2}(\R)$ is a one-to-one and
    onto linear transformation, then the standard matrix $A_T$ of $T$
    is invertible.
  \item If $S$ is a linearly independent set of vectors in a vector
    space $V$, then $S$ is a basis of $\Span S$.
  \item If $P(t)$ is an $n\times n$ matrix whose columns are solutions
    to the system of differential equations $\vec{x}'(t)=A\vec{x}(t)$,
    then $P(t)$ is a fundamental matrix of the system.
  \item If $A$ is $3\times 3$ with a one-dimensional nullspace, then
    the matrix $A$ has dimension $2$.
  \end{itemize}

\item True/False (proof or counterexample)
  \begin{itemize}
  \item If $A,B,C$ are symmetrix $n\times n$ matrices, then $AB+C$ is
    symmetric.
  \item If $A,B$ are matrices such that $AB=I_n$, then both $A,B$ are
    invertible.
  \item If $S=\{v_1,\dots,v_n\}$ is a set of vectors in some vector
    space, and some linear combination of them is $\vec{0}$, then $S$
    is a linearly independent set.
  \item If $A$ is a matrix, then $\Row A=\Row A^TA$.
  \item The initial value problem $ay''+by'+cy=0$, $y(0)=0$ has a
    unique solution for every $a,b,c\in\R$.
  \item If $A$ is $n\times n$ and $A^2=0$, then the characteristic
    polynomial of $A$ is $\lambda^n$.
  \item If $A$ can be row-reduced to $B$, then $A$ and $B$ have the
    same determinant.
  \item If $A$ is $5\times 6$ and $\dim\Nul A=1$, then
    $T(\vec{x})=A\vec{x}$ is onto.
  \item Every linearly independent subset of a finite-dimensional
    vector space $V$ has a subset which spans $V$.
  \item If $\{f_1,f_2,f_3\}$ is a linearly independent set of
    functions from $\R$ to $\R$, then the Wronskian
    $W[f_1,f_2,f_3](t)\neq 0$ for all $t$.
  \item If $\{v_1,\dots,v_n\}$ is a set of nonzero orthogonal vectors
    then $\{v_1,\dots,v_n\}$ is linearly independent.
  \item Linearly independent eigenvectors have distinct eigenvalues.
  \end{itemize}
\item Find a fundamental matrix for a system $\vec{x}'(t)=A\vec{x}$
  and use it to compute $e^{tA}$.

\item Find the general solution to the differential equation
  $y''(x)-y'(x)-2y(x)=\cos x-\sin 2x$.

\item Give an example of a matrix $A$ and a vector $\vec{b}$ such that
  $A\vec{x}=\vec{b}$ has a non-unique solution.
\item Find all solutions to the ODE $y''-4y'+5y=te^{2t}$ such that $y(0)=y'(0)$.
\item A $3\times 3$ matrix contains integer entries, where the
  diagonal entries are odd and the non-diagonal entries are even.  Is
  the matrix always invertible, sometimes invertible, or never
  invertible?

\item Let $A=(0,1,1;1,0,1;1,1,0)$. Find the eigenvalues, find bases
  for the eigenspaces, orthogonally diagonalize $A$.
\item Let $A$ be a square matrix, $\lambda$ a real eigenvalue with
  eigenvector $\vec{u}$. Prove that the vector-valued function
  $\vec{x}(t)=e^{\lambda t}\vec{u}$ is a solution to the system
  $\vec{x}'=A\vec{x}$.
\item Suppose that $A$ is a $4\times 4$ matrix. Suppose that
  $v_1,v_2,v_3$ are nonzero vectors in $\R^4$ such that
  $Av_1=(1,0,0,0)$, $Av_2=(0,-2,0,0)$, $Av_3=(0,0,0,0)$. Determine the
  minimum and maxmimum possible rank of $A$.
\item Suppose that $T$ is an $n\times n$ matrix satisfying
  $T^2=-I_n$. Prove that $n$ must be even.
\item Suppose that a square matrix $A$ satisfies $(A-\lambda I)^k=0$
  for some positive integer $k$. Prove that the only eigenvalue of $A$
  is $\lambda$.
\item Given a matrix, compute $e^{tB}$.

\item Is a given matrix diagonalizable?
\item Suppose $A$ is a $3\times 3$ matrix with a basis of orthonormal
  eigenvectors $\{v_1,v_2,v_3\}$. Let $Q=[v_1\quad v_2\quad
  v_3]$. Which of the following is always diagonal? (a) $Q^TAQ$ (b)
  $QAQ^T$ (c) $QAQ^{-1}$ (d) None of these.
\item Given particular solutions for different equations with the same
  homogeneous solution, superposition.
\item Solve the initial value problem $x'(t)=(1,1;0,1)x(t)+(e^{2t},0)$, $x(0)=(1,0)$.

\item For a matrix $A$, find a matrix $B$ such that $B^2=A$.