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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 3, due Wed Oct 5}

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

\vskip .05in

\ben

% 7 pts
\item 

Let $T$ be a linear transformation given by $T(x_1,x_2)=\left(\frac32
x_1+\frac12 x_2, x_1-x_2, x_2\right)$.
\ben
\item Find the standard matrix $A$ for the transformation $T$:

\vspace{1.5in}

\item Find a vector $\boldsymbol{x}$ whose image under $T$ is $(5, 2, 1)$:

\vspace{2.5in}

\item What is the definition for a linear transformation $T\,:\,
\mathbb{R}^n\rightarrow\mathbb{R}^m$ to be one-to-one?

\vspace{1.5in}

\item Is this transformation $T$ one-to-one? Justify your answer.
\een


\newpage %%%%%%%%%%%%%%%%
% 7 pts
\item
Let $A = \left[\begin{array}{lll}0 & 1 & 0\\1& 2& 0\\ 2& 4& 1\end{array}
\right]$
\ben
\item If $A$ is not invertible,
prove it, else, if $A$ is invertible, find $A^{-1}$:

\vspace{3in}

\item Use part (a) to express the vector $(4, 1, -1)$ as a linear
combination of the columns $\mbf{a}_1$, $\mbf{a}_2$, $\mbf{a}_3$ of $A$:

\vspace{2in}

\item Now let $A$ be a general invertible $n\times n$ matrix.  Prove
  that $A^T$ is invertible. [Note: use only the definition of
  invertible and properties of the transpose.  Make sure every
  object you use exists! Do not use the IMT $\stackrel{\cdot\;\cdot}{\smile}$ ]


\een

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

% 6 pts
\item
An alien species has two forms: ``larval'' and ``flying.''
Each year half the larval turn into flying ones, with the other half
staying larval.
Also, each year, 10\% of the flying ones die, and the others stay flying.
\ben
\item Stacking the larval and flying populations into a column vector $\mbf{x}$,
write the population vector at year $k+1$ in terms of that at year $k$.
Be sure to give the matrix.

\vspace{3in}

\item If the populations at ``year zero'' (the year the aliens land on
Earth\ldots) are 200 larval and 0 flying, what are the populations
at year 3?

\vspace{3in}

\item[BONUS:] What is the long-term fate of the alien race,
i.e.\ the vector $\lim_{k\to\infty} \mbf{x}^{(k)}$ ?
\een

\een

\end{document}
