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\begin{document}

Your name:
\vskip 12pt

Instructor (please circle): 
\hspace{1in}
Alex Barnett
\hskip .7in
Naomi Tanabe
\vskip 12pt

{\bf \large Math 22 Fall 2016, Homework 6, due Wed Oct 26}

{\em Please show your work. No credit is given for solutions without work or justification.}

\vskip .05in

\ben

% 7 pts
\item
\ben
\item Let 
%$A=\bma 0& 1& 1 & 1\\ 1&2&-1&0\\-2&0&6&4\ema$.
$A=\bma 0&1&-2\\ 0 &-2&4\\1&-1&6\\1&0&4\ema$.
Find a basis for Row $A$:

\vspace{2.5in}

\item
For the previous $A$, find a nontrivial vector that is guaranteed to have a
zero dot product with every vector in Row $A$. 
In what fundamental subspace for the matrix $A$ does such a vector live?

\vspace{1.5in}

\item What are the possible
ranks of some $5\times 3$ matrix $B$ if there exists two nontrivial solutions to
$B\mbf{x} =\mbf{0}$, neither of which is a multiple of the other?

\vspace{1in}

\item For a general matrix $A$, is the set Col $A$ affected by row reductions?

\vspace{0.5in}

\item[BONUS.] Is the dimension of Col $A$ affected by row reductions?


\een



\newpage %%%%%%%%%%%%%%%%

% 7 pts
\item 
\ben
\item  For what $h$ does the matrix $\bma 11 & 1\\h & 3\ema$
have an eigenvalue of (algebraic) multiplicity two? Show your work, as usual.

\vspace{2in}

\item For this $h$ in the above matrix, what is the dimension of the eigenspace?

\vspace{1.5in}

\item For this $h$ in the matrix from (a), explain why there can be no other eigenvalues.

\vspace{1in}

\item The vector $\mbf{v} = (0,0,1)$ is an eigenvector of
$\bma -1&-4&0\\2&5&0\\4&4&3\ema$.
Find another vector that together with $\mbf{v}$ forms a basis
for the eigenspace in which $\mbf{v}$ lives:




\een




\newpage %%%%%%%%%%%%%%%%

% 6 pts
\item
Find all eigenvalues of the matrix $A = \bma 1&1&-1\\-5&3&-3 \\ -4 & 0& 0\ema$.
For each eigenvalue state its (algebraic) multiplicity,
give a basis for its eigenspace and state the dimension of the eigenspace.
[Hint: by noticing something you already know about the matrix, you can
know one eigenvalue immediately. This helps you check your work.]






\een

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