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\pagestyle{myheadings} \markright{Math 4310 (Fall 2020) \hspace{2in}
(HW6)}

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  {\large Homework \# 6 and \#7 ({\color{red}revised})

    \medskip
    Math 4310 Fall 2020

    \medskip
    
    Due Friday, 10/30/20
  }

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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

The first 4 problems form HW6 (these are about quotient vector
spaces), and the last 4 problems form HW7 (these are about dual vector
spaces and dual linear maps).

\begin{enumerate}

\item Suppose that $T : V \longrightarrow W$ is a linear map ($V$ and $W$ are vector spaces, not necessarily finite dimensional).
  Define a new map $\widetilde{T} : V/\ker T \longrightarrow W$, defined as
  $\widetilde{T}({\bf v} + \ker T) = T({\bf v})$.
  \babc
\item Prove that $\widetilde{T}$ is well defined.
\item Prove that $\widetilde{T}$ is a linear map.
\item Prive that $\ker \widetilde{T} = 0$.
\item Prove that if $T$ is surjective, then $\widetilde{T}$ is an isomorphism.
  ({\color{red} This is a good way to get isomorphisms of quotient vector spaces with something more ``friendly''!}).
  
\item Suppose that $V = U \oplus W$, where $U$ and $W$ are subspaces of $V$.  Consider the linear map
  $T : V \longrightarrow W$, which sends ${\bf u} + {\bf w}$ to $T({\bf u} + {\bf w}) = {\bf w}$.
  What can you say about $\widetilde{T}$ for this example? (You may use in later problems what you learn in this problem!)
  \eabc
  
\item Let $V$ be a vector space over the field $\F$, and let $W$ be a subspace.  Recall that
  a {\em complement} of $W$ is a vector subspace of $V$ such that $V = U \oplus W$ (and we call $U$ and $W$ complementary).

  If $U_1$ and $U_2$ are both complements of $W$ in $V$:
  \babc
\item Show that $U_1$ and $U_2$ are isomorphic.
\item Is $U_1 = U_2$?  Either prove this, or give a counter-example.
\item Let $O(\R) = \{ f\in C^\infty(\R)\ | \ f(-x) = -f(x)\}$ be the set of {\bf odd} smooth functions and let $E(\R) = \{ f\in C^\infty(\R)\ | \ f(-x) = f(x)\}$ be the
set of {\bf even} smooth functions.  Prove that $O(\R)$ and $E(\R)$ are complementary. 
Prove that $C^\infty(\R)/O(\R) \cong E(\R)$.
  \eabc


\item Let $U$ and $V$ be vector spaces with respective subspaces $X$ and $Y$.  Prove that there is an isomorphism 
$(U\times V)/(X\times Y)\cong (U/X)\times (V/Y)$.

  
\item Let $C^\infty(\R)$ denote the vector space (over $\R$) of infinitely-differentiable real-valued functions $f:\R\to \R$.
(Note: $C^\infty(\R)$ is {\bf very} infinite-dimensional!)

Let $W$ denote the subspace of $C^\infty(\R)$ consisting of those functions which ``vanish to $n^{\mathrm{th}}$ order
at $0$":
$$
W = \left\{ f\in C^\infty(\R)\ \bigg| \ f(0) = 0,\ \frac{df}{dx}(0)=0, \dots,  \mbox{ and } \frac{d^nf}{dx^n}(0)=0 \right\}.
$$
Prove that the quotient vector space $C^\infty(\R)/W$ is finite-dimensional and find a basis.

\item Suppose that $m$ is a positive integer, and let $V = \PP_m(\R)$ be the vector space of
  polynomials of degree at most $m$.  Consider the basis $\mathcal{A} = (1, x, x^2, \ldots, x^m)$ of $V$.
  \babc
  \item Show that
    the dual basis to $\mathcal{A}$ is $\mathcal{A}^* = (\phi_0, \phi_1, \ldots, \phi_m)$, where
    \[ \phi_j(p(x)) = \frac{p^{(j)}(0)}{j!}. \]
  \item Show that $\mathcal{B} = (1, x-3, (x-3)^2, \ldots, (x-3)^m)$ is a basis of $V$.
    \item Find the dual basis $\mathcal{B}^*$.
      \eabc

    \item Suppose that $\mathcal{A} = ({\bf v}_1, \ldots, {\bf v}_n)$ is a basis for a vector space $V$, and
      that
      $\mathcal{B} = ({\bf w}_1, \ldots, {\bf w}_m)$ is a basis for a vector space $W$.  Suppose that
      $T : V \longrightarrow W$ is a linear map.  Also suppose that the dual bases of $V^*$ and $W^*$
      are (respectively) $\mathcal{A}^*$, $\mathcal{B}^*$.
      If $A = (A_{i,j})$ is the matrix $[T]_{{\mathcal B} \leftarrow {\mathcal A}}$, then show that the
      matrix of $T^* : W^* \longrightarrow V^*$ (i.e. $[T^*]_{{\mathcal A}^* \leftarrow {\mathcal B}^*}$), is the transpose of $A$.

    \item {\color{red} Assume that $V$ is a finite dimensional vector space}.
      Recall that if $U \subset V$ is a subspace of $V$, then
      \[ U^o := \{ \varphi \in V^* \mid \varphi(u) = 0 \mbox{\ for all $u \in U$} \}. \]
      \babc
    \item The inclusion map $i : U \longrightarrow V$ is the linear map
      sending ${\bf u} \in U$ to ${\bf u}$ (but in $V$).
      By using the fundamental theorem of linear algebra on the dual of this map,
      show that $\dim U + \dim U^o = \dim V$.
      \item For which $U \subseteq V$ is $U^o = 0$?  For which $U$ is $U^o = {\color{red} V^*}$? (State a result, and prove it).
\eabc

    \item {\color{red} Assume that $V$ and $W$ are finite dimensional vector spaces}.
      Let $T : V \longrightarrow W$ be a linear map with dual $T^* : W^* \longrightarrow V^*$.
      (In class, we used the notation $T^t$, but I like the notation $T^*$ better).  Recall that $T^*(\varphi) = \varphi T$,
      where $\varphi : W \longrightarrow \F$.
      \babc
    \item Show that $\ker(T^*) = (image(T))^o$.
    \item Show that $\dim image(T) = \dim image(T^*)$
    \item Show that $image(T^*) \subset (\ker(T))^o$.
      Show that $image(T^*)$ and $(\ker(T))^o$ have the same dimension, and therefore that
      these subspaces of $V^*$ are the same.
    \item Show that $T^*$ is surjective if and only if $T$ is injective.
      \item Show that $T^*$ is injective if and only if $T$ is surjective.
      \eabc
      \pagebreak
    \item
      {\color{red} This problem is {\bf being moved} to the {\bf next} homework set!  I include it here,
        but it is not due this week!}.
      Suppose that $A \in \F^{m \times n}$, and let $L_A : \F^n \longrightarrow \F^m$ be the corresponding
      linear map. We often represent elements of $\F^n$ as column vectors.
      {\color{red} Elements of ${\F^n}^* = \LL(\F^n, \F^1)$ can be thought of as $1 \times n$ matrices, i.e. row vectors}.

      {\color{red} Previous description, not as clear: } Similarly,
      we often represent elements of ${\F^n}^*$ as row vectors, where the row vector $e_i^*$ (with a 1 in the
      $i$th column, zeros elsewhere, corresponds to $\varphi_i \in V^*$. 

      A matrix $A$ has 4 interesting subspaces
      associated with it:
      \babc
      \item[(i)] $colspan(A) \subset \F^m$ is the span of the columns of $A$,
      \item[(ii)] $rowspan(A) \subset {\color{red}\F^n}^*$ is the span of the rows of $A$,
      \item[(iii)] $\ker(A) = \{ {\bf x} \mid           A{\bf x} = {\bf 0} \}$,
        \item[(iv)]
          the left kernel of $A$,
            $leftker(A) = \{ {\bf y} \in {\F^m}^* \mid {\bf y} A = {\bf 0}\}$.
            \eabc

      \noindent
      For this problem, suppose that $T = L_A$.  
            \babc
          \item If ${\bf w} = (a_1, \ldots, a_n)$ is a row vector (i.e. a $n \times 1$ matrix)
            {\color{red} representing an element of ${\F^n}^* = \LL(\F^n, \F^1)$,}
%              which represents the element
%              $\varphi = a_1 \varphi_1 + \ldots + a_n \varphi_n \in {\F^n}^*$,
              and
            if ${\bf v} = b_1 e_1 + \ldots + b_n e_n \in \F^n$ is a column vector,
            what is $\varphi({\bf v})$?
          \item In the previous problem, several subspaces are mentioned:
            \[\ker(T^*), \ker(T), image(T), image(T^*), (\ker(T))^o, (image(T))^o. \]
            Some of these are the same as the 4 subspaces associated to $A$ above.
            Match them up, and prove that they are the same.
      \eabc


\end{enumerate}

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