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\pagestyle{myheadings} \markright{Math 4310 (Fall 2020) \hspace{2in}
(HW4)}

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  {\large {\color{red} Revised} Homework \# 4 $\quad$ Math 4310 Fall 2020

    \medskip
    
    Due 9/30/20
  }

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Please submit your completed homework via gradescope on canvas.
I encourage you to work with your classmates on this homework (except for the journal entries!)
When you submit your work, please {\bf list your collaborators}.
(Your grade will not be affected.)
Even if you work in a group, you should write up 
your solutions {\bf yourself}! 
You should include all computational details, and proofs should be carefully 
written with full details.
As always, please write {\bf neatly and legibly} (feel free to use \LaTeX\ to write up your solutions, if you wish!).

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\noindent {\bf Journal entry.}
There is no journal entry this week.

\vskip 0.1in

\noindent {\bf Exercises.} 

\vskip 0.1in

\begin{enumerate}

    \item
      \babc
\item If $\F = \Z_p$, where $p$ is a prime number, and $V$ is a vector space of dimension $n$
  over $\F$,  how many elements does $V$ have?
\item How many (ordered) bases do the vector spaces $\F^1$ and $\F^2$ have, if $\F = \Z_2$?  if $\F = \Z_p?$.
\item How many (ordered) bases does the vector space $\F^3$ have, if $\F = \Z_2$?  if $\F = \Z_p?$.
\eabc

\item Given a vector space $V$ over a field $\F$, we define the {\bf dual vector space} $V^* := \LL(V, \F)$.
  If $V$ is finite dimensional of dimension $n$, find the dimension of $V^*$, and find a basis of this
  vector space.

\item Find the dimensions of the following vector spaces.  Suppose that $\dim U = a$, $\dim V = b$, and $\dim W = c$.
  (Recall that for this course, you must always give reasons for your answers, simply writing down the dimension is not
  good enough!) {\color{red} You may use the fact that $\dim \LL(V,W) = \dim V \dim W$ without proof in this problem.
    All sums here are direct products (aka external direct sums), not our original notion of direct sum!}
  \babc
\item $U \times V$ (note: we often write this as $U \oplus V$, even though $U$ and $V$ might not be subspaces
  of a common vector space.  We do this below too).
  \item $\LL(V, U)$.
  \item $\LL(V, U \oplus U)$.
%  \item $\LL(\PP_2(\F), U \oplus U)$.
  \item $U \oplus V \oplus W$.
    \item $\mathscr{F}un({\color{red}[2]} \times [2] \times [2], \F)$, where $[n] = \{ 1, 2, \ldots, n\}$. 
  \eabc

\item Find bases for both the kernel and image of the following maps, together with their dimensions.  Which
  is injective, which is surjective?  {\color{red} You may assume that these are all linear maps}.
  \babc
\item $T : \PP_2(\R) \longrightarrow \PP_3(\R)$, where $T(f(x)) := x f(x) + f'(x)$.
\item $T : \R^{2 \times 2} \longrightarrow \R^{2 \times 2}$ defined by
  $T(A) := AB - BA$, where $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.
\item $D : \PP(\R) \longrightarrow \PP(\R)$, defined by $D(f(x)) = f'(x)$.
  \eabc
  
  \item Let $V$ be a finite dimensional vector space over a field $\F$, let $X$ be a non-empty finite set, and
let $\mathscr{F}\mathrm{un}(X,V)$ denote the vector space of functions from the set $X$ to the vector space $V$.
(You do not need to prove that this is a vector space over $\F$.)
Fix an element $a\in X$, and 
define a function
\begin{eqnarray*}
T_a: \mathscr{F}un(X,V) & \to & V\\
f & \mapsto & f(a).
\end{eqnarray*}
Prove that $T_a$ is a linear transformation, and find $\ker(T_a)$ and $Im(T_a)$.  What are the dimensions of
$\mathscr{F}\mathrm{un}(X,V)$, $\ker(T_a)$ and $Im(T_a)$?
{\color{red} Finding $\ker(T_a)$ does not mean to necessarily find a basis.  You can also describe it as all functions
  that satisfy certain properties.
  Of course, you can use bases too, if you show they are bases}.
  

\item Let $T : \F \longrightarrow \F$ be defined to be $T(x) = x^3$.
  \babc
\item Is this a linear map if $\F = \Z_2$? (Either prove, or show why not).
\item Is this a linear map if $\F = \Z_3$? (Either prove, or show why not).
\item Is this a linear map if $\F$ has characteristic not equal to 3? (Either prove, or show why not).

  \eabc

\item Recall that if $z = a + bi \in \C$, where $a, b$ are real numbers, then its conjugate is $\bar{z} := a - bi$.
  Consider the function $f : \C \longrightarrow \C$, where $f(z) = \bar{z}$.
  \babc
  \item If we consider this a map of $\C$-vector spaces, is $f$ a linear map?
    \item If we consider this a map of $\R$-vector spaces, is $f$ a linear map?
  \eabc
%% \item For an $m \times n$ matrix $A$, over a field $\F$, let $L_A : \F^n \longrightarrow \F^m$ denote the
%%   corresponding linear map (i.e. $L_A({\bf v}) := A {\bf v} \in \F^m$).
%%   Prove that 


\end{enumerate}

\end{document}

