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\pagestyle{myheadings} \markright{Math 4310 (Fall 2020) \hspace{2in}
(HW2)}

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  {\large Homework \# 2 $\quad$ Math 4310 Fall 2020

    \medskip
    
    Due 9/16/20
  }

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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!).

\bigskip 
In the exercises below (except for the first 2 problems),
you may use any theorem or proposition stated in class, or in the corresponding sections
of Axler's book.  But, make sure you state carefully what results you are using!  (e.g.
don't say ``by a theorem we proved in class, ...'' (without stating the result!)).


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

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\noindent {\bf Exercises.} 

\vskip 0.1in

\begin{enumerate}

  \item Let $\F$ be a field.  Prove from the axioms/properties:
  \babc
\item (Cancellation) For all $a, b, c \in F$, then (a) $a + b = a + c$ implies that $b = c$,
  and (b) if $a \ne 0_\F$, and $ab = ac$, then $b = c$.
\item State and prove a lemma that the additive identity and additive inverse are uniquely defined.
\item Show that for all $a \in \F$, $0_\F \cdot a = 0$.
  \item Show that $(-a) = (-1) \cdot a$, for all $a \in \F$.
\item Show that for all $a, b \in \F$, that $(-a) \cdot (-b) = a \cdot b$.
  \eabc

\item Let $V$ be a vector space over $\F$.  Prove from the axioms/properties (you may also use the
  statements in the previous problem):
  \babc
\item (Cancellation) For all $u, v, w \in V$, if $u+v = u + w$, then $v = w$.
\item For all $a \in \F$, and $u, v \in V$, if $a u = a v$, then either $a = {\mathbb 0}_\F$ or $u = v$.
\item For all $a,b \in \F$, and $u \in V$, if $a u = b u$, then either $a = b$ or $u = {\mathbb 0}_V$.
\item ${\mathbb 0}_\F \cdot v = {\mathbb 0}_V$.
\item If $c \in \F$, then $c \cdot {\mathbb 0}_V = {\mathbb 0}_V$.
  \item $(-1) v = - v$, for all $v \in V$.
  \eabc

  \eject
  
\item Find the reduced row echelon form for the following matrices over the given fields.
  \babc
\item $\begin{pmatrix}
  1 & 0 & 0 & 1 \\
  1 & 1 & 1 & 0 \\
  0 & 1 & 1 & 1
\end{pmatrix}$, over the field $\Z_2$.
\item $\begin{pmatrix}
  1 & 3 & 2 & 0 & 1 \\
  3 & 2 & 5 & 1 & 0 \\
  0 & 0 & 5 & 2 & 1
\end{pmatrix}$, over the field $\Z_7$.

\item $\begin{pmatrix}
  1+i  & 1-i & 0 \\
  2i &   2   & 1 \\
  2  &  -2i   & -1
\end{pmatrix}$, over the field $\C$.
  
  \eabc

\item Determine whether the following subsets form vector subspaces (using the
  induced addition and scalar multiplication from the larger vector space).
  \babc
  \item $W = \{ p(x) \in \mathcal{P} \mid p''(2) = 0 \} $
  \item $W = \{ p(x) \in \mathcal{P} \mid p'(0) = 2 \} $
  \item The set of circulant $4 \times 4$ matrices.
    These are the matrices which can be written
    \[\begin{pmatrix}
      a_0 & a_1 & a_2 & a_3 \\
      a_3 & a_0 & a_1 & a_2 \\
      a_2 & a_3 & a_0 & a_1 \\
      a_1 & a_2 & a_3 & a_0
    \end{pmatrix}
    \]
    for some $a_0, a_1, a_2, a_3 \in \F$.
    \item $W = \{ A \in \R^{2 \times 2} \mid A^2 = A \}$.
  \eabc

\item
\babc
\item Show that $\C$ is a vector space over $\R$ (with the ``usual'' operations).
\item Consider the field $\F_4$ that you constructed in the first homework.
  Let $\Z_2$ be the subset consisting of the 2 elements $0, 1$.
  Show that $F_4$ is a vector space over $\Z_2$.
\item In general, if $\F \subset L$, and $\F$ and $L$ are fields (where $\F$ is a
  field using the induced operations from $L$), then show that $L$ is a vector space over
  $\F$.
  \eabc

\item Let $V$ and $W$ be vector spaces over $\F$.  Let $V \times W$ be the set of all
  pairs \[\{ ({\bf v}, {\bf w}) \mid {\bf v} \in V, {\bf w} \in W \}.\]
  Define operations (and the zero element) which makes $V \times W$ into a
  vector space over $\F$.
  (If you choose the operations I think you should, this is called the (external)
  direct product of $V$ and $W$).

  \item Let $V=\Z_4$ be the integers modulo $4$ (as defined in class).  Is it possible to define the structure of a vector space on $V$ over
$\F_2$ so that vector addition is the addition we defined for $\Z_4$?

    \eject
\item Let $\R^+$ denote the positive real numbers.  Let
$$
V = \left\{ \left(\begin{array}{c}a\\ b\end{array}\right)\ \Big|\ a,b\in\R^+ \ \right\},
$$
and define vector addition on $V$ by $\left(\begin{array}{c}a\\ b\end{array}\right)+ \left(\begin{array}{c}c\\ d\end{array}\right)=
\left(\begin{array}{c}ac \\ bd\end{array}\right)$ and scalar multiplication by $e\in\R$ to be
$e\cdot \left(\begin{array}{c}a\\ b\end{array}\right) = \left(\begin{array}{c}a^e\\ b^e \end{array}\right)$.
Prove that $V$ is a vector space over $\R$.

\end{enumerate}

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