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\noindent
\parbox{2.25in}{\bf Math 3510}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Oct. $8^\mathrm{th}$, 2018}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Dual Basis] Let $\alpha = \{ (1,0,0), (1,-1,0), (2,0,1) \}$. 
\begin{enumerate}[(a)]
\item Explain why $\alpha$ is a basis for $\mathbb{R}^3$. 
\item Find $\alpha^*$ for $(\mathbb{R}^3)^*$ (i.e. find the basis dual to $\alpha$). 

{\it Note:} By ``find'' I mean give formulas for dual vectors like the formula given for $f$ in part (c).
\item Explain why $f \in (\mathbb{R}^3)^*$ where $f(x,y,z)=3x+2y+z$ (what is the definition of a dual vector?). Then write $f$ as a linear combination of $\alpha^*$ elements (i.e. find its $\alpha^*$-coordinates).
\end{enumerate}

\item[\#2 Dual Proof] Let $W$ be a subspace of a vector space $V$ (over a field $\mathbb{F}$). We say that $f \in V^*$ {\bf annihilates} $W$ if $f(\B w)=0$ for all $\B w \in W$. Let $A(W) = \{ f \in V^* \;|\; f(\B w) = 0 \mbox{ for all } \B w \in W \}$ (the collection of all linear functionals which annihilate $W$). $A(W)$ is called the {\bf annihilator} of $W$. 

\begin{enumerate}[(a)]
\item Prove that $A(W)$ is a subspace of $V^*$.
\item {}{\bf [Optional:]} Suppose that $V=U \oplus W$ for some subspaces $U$ and $W$. Show that $V^* = A(W) \oplus A(U)$.
\item {}{\bf [Optional:]} Let $T:V \to V$ be a linear operator and suppose that $T(W) \subseteq W$ for some subspace $W$ (i.e. $W$ is a $T$-invariant subspace).  
          Show that $T^t(A(W)) \subseteq A(W)$ (i.e. $A(W)$ is $T^t$-invariant).
\end{enumerate}

\item[\#3 Notational Issues] Recall that according to Einstein's summation convention, the simultaneous appearance of an upper and lower index implies a summation: $a_{imn}{}^{jk}b_{jk}{}^{m\ell} = \sum\limits_{m}\sum\limits_{j}\sum\limits_{k} a_{imn}{}^{jk}b_{jk}{}^{m\ell} = c_{in}{}^{\ell}$.

The Levi-Civita symbol is a close companion of the Kronecker delta. The Levi-Civita symbol on 2 indices is defined by $\epsilon_{11}=\epsilon_{22}=0$, $\epsilon_{12}=1$, and $\epsilon_{21}=-1$. On 3 indices the symbol is defined by (for $i,j,k \in \{1,2,3\}$)\dots
$$\epsilon_{ijk}  = \left\{ \begin{array}{ccl} +1 & & (i,j,k)=(1,2,3),\; (2,3,1),\; (3,1,2) \\
                                                                         -1 & & (i,j,k)=(1,3,2),\; (2,1,3),\; (3,2,1) \\
                                                                          0 & & i=j \mbox{ or } j=k \mbox{ or } i=k \end{array} \right.$$
Notice that $(1,3,2)$ can be obtained from $(1,2,3)$ after a single interchange: $2^\mathrm{nd} \leftrightarrow 3^\mathrm{rd}$. The same is true for $(2,1,3)$ and $(3,2,1)$. In particular, $(1,3,2)$, $(2,1,3)$, and $(3,2,1)$ are all {\it odd} permutations of $(1,2,3)$ (we get from the identity $(1,2,3)$ to our triple after an odd number of interchanges). 

On the other hand, $(2,3,1)$ requires two interchanges. From $(1,2,3)$ we interchange $1^\mathrm{st} \leftrightarrow 2^\mathrm{nd}$ and get $(2,1,3)$ then interchange $2^\mathrm{nd} \leftrightarrow 3^\mathrm{rd}$ and get $(2,3,1)$. The same is true for $(3,2,1)$. Finally, $(1,2,3)$ requires no interchanges at all. In particular, $(1,2,3)$, $(2,1,3)$, and $(3,2,1)$ are all {\it even} permutations of $(1,2,3)$ (we get from the identity to our triple after an even number of interchanges).

More generally, $\epsilon_{i_1i_2\dots i_n}$ is defined to be $+1$ if $(i_1,\dots,i_n)$ is an even permutation of $(1,\dots,n)$. It's $-1$ for an odd permutation and $0$ if $i_k=i_\ell$ for some $k \not= \ell$ (there's a repeated index).

{\bf WARNING:} We are using Einstein's summation convention in this problem.

\begin{enumerate}[(a)]
\item Assuming $a_{ijk}{}^{\ell m}$ and $b_{x}{}^{yz}$ are indexed collections of scalars where $i,j,k,\ell,m,x,y,z \in \{1,2,\dots,n\}$.
        Identify the appropriate sub/super-scripts on $c$ if $a_{ijk}{}^{i\ell}b_{\ell}{}^{jx} = c_{???}{}^{???}$.  What about $a_{ijk}{}^{\ell j} b_{\ell}{}^{ki}$?

\item Let $A$ be a $2 \times 2$ matrix with entries $A_i{}^j$. What does $\epsilon_{ij}A_1{}^iA_2{}^j$ compute? (Write this out explicitly and identify this as a familiar formula: $\epsilon_{ij}A_1{}^iA_2{}^j$ is the ??? of $A$.)

         More generally, for a $3 \times 3$ matrix, what does $\epsilon_{ijk}A_1{}^{i}A_2{}^{j}A_3{}^{k}$ compute? Or for an $n \times n$ matrix, what does $\epsilon_{i_1\dots i_n}A_1{}^{i_1}\cdots A_n{}^{i_n}$ compute? [No proof/calculation necessary. Just identify what these expression compute.]
         
\item Let $A$ be an $n \times n$ matrix with entries $A_i{}^j$. What does $A_i{}^i$ compute?

\end{enumerate}

\end{description}

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