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\pagestyle{myheadings} \markright{Math 4310 (Fall 2020) \hspace{2in}
(HW3)}

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  {\large Homework \# 3 $\quad$ Math 4310 Fall 2020

    \medskip
    
    Due 9/23/20
  }

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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.

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\noindent {\bf Exercises.} 

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\begin{enumerate}

\item Find bases and the dimension of the following vector spaces (Make sure you believe that they are subspaces, but you do not need to
  prove that here!).
  \babc
\item $W = \{ A \in \Q^{2 \times 2} \mid AB = BA \}$, where $B = \begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}$.
\item $U = \{ {\bf v} \in (\Z_5)^4 \mid C {\bf v} = 0 \}$, where
  \[ C = \begin{pmatrix}
    1 & 3 & 2 & 0 \\
    4 & 2 & 4 & 1 
  \end{pmatrix}
  \]
  
  \eabc
      
    \item
      \babc
      \item Consider $\mathscr{P}_2(\R)$ of polynomials over $\R$ of degree at most 2.  Find polynomials $f_0$, $f_1$, $f_{-1}$
      which satisfy the following conditions: $f_0(0) = 1$, $f_0(1) = f_0(-1) = 0$, $f_1(1) = 1, f_1(-1) = f_1(0) = 0$, and
      $f_{-1}(-1) = 1, f_{-1}(0) = 0, f_{-1}(1) = 0$.  Is $(f_0, f_1, f_{-1})$ a basis of $\mathscr{P}_2(\R)$?  As always, prove your answer.
    \item The Bernstein polynomials for degree $n$ are $b_i = {n \choose i} x^i (1-x)^{n-i}$, for $i = 0, \ldots, n$.  Note that we
      always assume that the binomial coefficient ${a \choose 0} = 1$, for $a > 0$.  For example, for $n = 2$, we have
      $(1-x)^2, 2x(1-x), x^2 $.  Prove that these three polynomials form a basis of $\mathscr{P}_2(\R)$.
      Challenge (definitely optional!): prove that the $n+1$ Bernstein polynomials of degree $n$ form a basis of $\mathscr{P}_n(\R)$.
      \eabc
      
    \item
A permutation on $[n] := \{1, 2, \ldots, n\}$ is a (re)-arrangement of these numbers.
For example, $(2, 3, 1, 4)$ is a permutation on $[4]$, as is $(1, 2, 3, 4)$.

  An $n \times n$  permutation matrix is an $n \times n$ matrix which has exactly one ``1'' in each row and in each column, and
  all other entries are zero.  
  \babc
  \item Find all $3 \times 3$ permutation matrices over $\R$. Let $V \subset \R^{3 \times 3}$ be the
    subspace spanned by these matrices.  Find a basis for this subspace.  What is its dimension?
    (To explore, but not for homework: what happens for $n=4$, or higher $n$?)
  \item Show that there is a 1-1 correspondence between permutations on $[n]$ and $n \times n$ permutation matrices.
    Show that there are $n!$ $n \times n$ permutation matrices.  
  \eabc
  
\item For this problem (and for any problems or exams later in the course), you may use the following integrals.  If $m$ and $n$ are
  positive integers, then 
  \[\int_{-\pi}^{\pi} \sin(m x) \sin(n x) \, dx = \begin{cases}0 & m \ne n \\ \pi & m = n\end{cases} \]
    \[\int_{-\pi}^{\pi} \cos(m x) \cos(n x) \, dx = \begin{cases}0 & m \ne n \\ \pi & m = n\end{cases} \]
      \[\int_{-\pi}^{\pi} \cos(m x) \sin(n x) \, dx = 0 \]
      Consider the subspace $V_r \subset \mathscr{F}un(\R, \R)$ spanned by the functions
      \[ V_r := span(1, \sin(x), \sin(2x), \ldots, \sin(rx), \cos(x), \cos(2x), \ldots, \cos(rx)) \]
      \babc
    \item Show that $(1, \sin(x), \cos(x))$ is linearly independent, and therefore a basis of $V_1$.
    \item Is $(1, \sin^2(x), \cos^2(x))$ linearly independent?  Prove or disprove.
    \item Show that $(1, \sin(x), \cos(x), \sin(2x), \cos(2x))$ is linearly independent, and therefore a basis of $V_2$.
      (Hint: remember that if you have two functions that are equal to each other, then multiplying both sides by the same function,
      integrating both (using definite integral) sides, differentiating both sides, evaluating both sides at the same value, gives two
      things which are still equal).
      \eabc

  \item In this problem, you may assume that
  the space of infinite sequences in $\F$,
  \[ \F^\infty = \left\{ (a_1, a_2, a_3, \ldots) \mid a_i \in \F, \mbox{for all $i > 0$} \right\} \]
  is also a vector space.  (Note: there is no convergence conditions on these sequences!)

  Define a subset
  \[ W = \{ (a_1, a_2, a_3, \ldots) \mid a_i = a_{i-1} + a_{i-2}, \mbox{for $i \ge 3$} \}. \]
  You may also assume that $W$ is a subspace,
  although you should check for yourself that it is one!
  \babc
\item Is $W$ finitely generated?  If so, find a basis for $W$.  What is its dimension?
\item Find a complementary space $U$ of $W$, i.e. a subspace $U$ such that $U \bigoplus W = \F^\infty$.
  \eabc

  \item Let $V$ be a finite-dimensional vector space over a
field $\F$, and let $U,W,X$ be subspaces.
\babc
\item If $U\cap W = 0$, prove that $\dim(U+W) = \dim U + \dim W$.
\item  In general, prove $\dim(U+W) = \dim U + \dim W - \dim(U\cap W)$.
\item If you're familiar with the ``inclusion-exclusion principle'', you might
guess that part (b) generalizes to the equality 
\eabc
\[ \dim(U+W+X) = \dim(U) + \dim(W) + \dim(X) 
		- \dim(U\cap W) - \dim(U\cap X) - \dim(W\cap X) + \dim(U\cap W\cap X). \]
\babc 
\item[] Provide a counterexample to show this ``equality'' can be false!

(Hint: You can take the field $\F$ to be $\R$ and the vector space $V$ to be $\R^2$).
\eabc

\end{enumerate}

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