\documentclass[minion]{homework}
\doclabel{Math F314: Linear Algebra}
\docdate{Homework 2 Supplement}
\usepackage{cmacros}
\newcommand{\vs}{\mathbf{s}}
\newcommand{\vx}{\mathbf{x}}
\newcommand{\vB}{\mathbf{B}}
\newcommand{\vb}{\mathbf{b}}
\newcommand{\vr}{\mathbf{r}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\vu}{\mathbf{u}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\vw}{\mathbf{w}}
\newcommand{\vzero}{\mathbf{0}}
\begin{document}
\begin{aproblems}
\aproblem  Suppose a rocket is shot from height $h=0$ meters
at time $t=0$ seconds.  The height of the rocket is known
at times $t=1$, $t=2$, $t=3$ and $t=4$ seconds. These 
heights are $h_1,\ldots,h_4$, measured in meters.
\begin{subproblems}
\item  Determine the average velocity of the rocket
on the time intervals $[0,1]$ and $[1,2]$.
\item Let $v_k$ denote the average vertical velocity on 
time interval $[k-1,k]$
and let $\mathbf v=(v_1,\ldots,v_4)$.
Find a matrix $A$ such that
\[
A\mathbf h = \mathbf v
\]
where $\mathbf h = (h_1,h_2,h_3,h_4)$.  [Note that
we are computing velocities from positions; multiplying by 
matrix $A$ is kind of like taking a derivative.]
\item Suppose you do not know $h_2$ but you know $h_1=100$ m and $v_2=300$ m/s.
What is $h_2$?
\item Suppose you do not know $h_1$ and $h_2$ but you know 
$v_1=150$ m/s and $v_2=300$ m/s.  What are the values of $h_1$ and $h_2$?
\item Find a matrix $S$ such that if $\mathbf v$ is known we can
determine $\mathbf h$ via
\[
\mathbf h = S\mathbf v.
\]
[Notice that multiplying by 
$S$ converts velocities back into positions.  It is a kind
of integration operation.]
\end{subproblems}

\aproblem
Consider the matrix
\[
A=\begin{bmatrix}   1 & -2&   5\\
   2&   1&   0\\
   1&   3&  -5
\end{bmatrix}.
\]
Let $\mathbf u$, $\mathbf v$ and $\mathbf w$ denote the
columns of $A$ in that order, so $A=[\mathbf u\; \mathbf v\; \mathbf w]$.
\begin{subproblems}
\item  Show that the columns of $A$ are linearly dependent.
\item  Find a nonzero vector $\mathbf n$ such that $A\mathbf x=\mathbf b$
has a solution $\mathbf x$ only if $\mathbf n\cdot \mathbf b=0$.
\item Is there a solution if $\mathbf b=(-1,3,5)$?  Why?
\item Find three different solutions of $A\mathbf x = \mathbf 0$.
\item Find three different solutions of $A\mathbf x = \mathbf b$
if $\mathbf b=(-1,3,4)$.
\end{subproblems}

\aproblem Find a $3\times 3$ matrix $A$ such that
\[
A\mathbf x = \mathbf b
\]
has a solution if and only if $\mathbf b$ is a multiple of $(1,-2,4)$.
That is, for this matrix, a solution exists only if $\mathbf b$ lies
on a certain line (not just a certain plane!)


\end{aproblems}

\end{document}
