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\begin{document}

Your name:
\vskip 12pt

Instructor (please circle): 
\hspace{1in}
Alex Barnett
\hskip .7in
Naomi Tanabe
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{\bf \large Math 22 Fall 2016, Homework 9, due 8pm Tues, Nov 15}

{\em This one is slightly shorter since you have 1 day less.}

\vskip .05in

\ben

% 8 pts
\item
Let $x_1$ be the intercept and $x_2$ be the slope for a 
general linear function $y(t) = x_1 + x_2 t$.
Find the least squares fit for this given
the data $(0,4)$, $(1,-2)$, and $(3,0)$, which are three points $(t,y)$ in
the plane.
Here's how to set up the linear system (you don't need to read Sec.~6.6
unless interested).
The first point says $x_1 + x_2.0 = 4$, the next says $x_1 + x_2.1 = -2$,
and the last says $x_1 + x_2.3 = 0$.
\ben 
\item Find the least squares solution vector(s)
$\hat{\mbf{x}} = \bma \hat x_1\\ \hat x_2 \ema$.
Is the solution unique?

\vspace{5in}

\item What is the solution error? (Pythagorean sum of the $y$-errors
between each data point and the line, ie your usual $\|\hat{\mbf{b}} - \mbf{b}\|$ error)

\vspace{1in}

\newpage

\item
Let $A$ be any matrix, possibly rectangular. Prove that if $A^TA$ is invertible,
then the columns of $A$ are linearly independent.
% 2 pts

\vspace{2in}

\een


% 8 pts ------------
\item
\ben
\item Prove that, for any $m\times n$ matrix $A$, the set of
nonzero eigenvalues of $A^T A$
is precisely the set of nonzero eigenvalues of $A A^T$. [Don't use the SVD. Hint: watch out for the case $A\mbf{v} = 0$ !]

\vspace{3in}

\item
Consider the matrix $A = \bma 1 & 1 & 0 \\ 1 & 0& 1\ema$.
Over all vectors $\mbf{x}$ in $\RR^3$ with $\|\mbf{x}\| = 1$, what
is the largest $\|A\mbf{x}\|$ can be? [Hint: easier if exploit (a)]

\vspace{3in}

\newpage

\item Compute by hand the full SVD of $A$, ie give $U$, $\Sigma$, and $V$.
[Hints: you might find it easier to exploit that $U$ is also the {\em right} singular
vectors for $A^T$. Find the third column of $V$ however you like.]



\een

\een

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