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\noindent
\parbox{2.25in}{\bf Math 4010-101}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Jan. $24^\mathrm{th}$, 2022}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force


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\begin{description}
\item[\#1.] For each of the following determine if it is a field. If so, explain why. If not, give a {\bf concrete} counterexample to show it fails to be a field. Also, for each of these determine its characteristic.
$$ \text{(a) \  \  } \mathbb{Z}[x] \text{ (polynomials with integer coeffs.)} \qquad 
     \text{(b) \  \  } \mathbb{Q}[\sqrt{3}]=\{a+b\sqrt{3} \;|\; a,b \in \mathbb{Q} \} \qquad
     \text{(c) \  \  } \mathbb{Z}_{10} \text{ (integers mod 10)}
$$

\item[\#2.] The field of complex numbers $\mathbb{C}$ is {\it isomorphic} to the collection of matrices $\displaystyle S = \left\{ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \;\middle|\; a,b \in \mathbb{R} \right\}$. Isomorphic here means that the arithmetic of $\mathbb{C}$ and $S$ match up perfectly under the mapping $a+bi \mapsto \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$. 

For example, $(a+bi)+(c+di) = (a+c)+(b+d)i$ just like $\begin{bmatrix} a & -b \\ b & a \end{bmatrix}+\begin{bmatrix} c & -d \\ d & c \end{bmatrix}=\begin{bmatrix} a+c & -(b+d) \\ b+d & a+c \end{bmatrix}$.

\begin{enumerate}[(a)]
\item Show that multiplication matches under this mapping.
\item Show that multiplicative inverses match under this mapping.
\item Recall that transposing a matrix means turnings rows into columns (and vice-versa). What does the operation of transposing a matrix in $S$ relate to in $\mathbb{C}$? 
\item What does taking the determinant of  a matrix in $S$ relate to in $\mathbb{C}$?
\end{enumerate}

{\it Note:} Technically part (c) and (d) are not features of our field isomorphism. These extra relationships show that more than just the field structure of $\mathbb{C}$ is encoded in $S$.

\item[\#3.] Consider $W=\{(x_1,x_2,x_3,x_4,x_5) \in \mathbb{R}^5 \;|\; x_1+x_2+x_3+x_4+x_5=0 \}$. 

\begin{enumerate}[(a)]
\item Explain why $W$ is a subspace of $\mathbb{R}^5$.
\item Verify that $S=\{(2,-3,4,-5,2), (-6,9,-12,15,-6), (3,-2,7,-9,1), (2,-8,2,-2,6),$\\ \force \hfill $(-1,1,2,1,-3), (0,-3,-18,9,12), (1,0,-2,3,-2), (2,-1,1,-9,7) \}$ is a subset of $W$.

For easy copy/paste data if you compute in Maple:

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{\color{red}
\verb#<<2,-3,4,-5,2>|<-6,9,-12,15,-6>|<3,-2,7,-9,1>|<2,-8,2,-2,6>|#\\
\verb#<-1,1,2,1,-3>|<0,-3,-18,9,12>|<1,0,-2,3,-2>|<2,-1,1,-9,7>>;#
}

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\item Justify why $S$ spans $W$ and find a linearly independent subset of $S$ (i.e., determine a basis for $W$ that is contained in $S$).

\item Write the remaining elements of $S$ as linear combinations of the basis elements you found in part (c).
\end{enumerate}

\item[\#4.] Let $\mathbb{F}$ be some field. For each of the following sets: Explain why either this is a subspace or why it fails to be a subspace. If it is a subspace, find a basis and compute its dimension.
\begin{enumerate}[(a)]
\item $W_1 = \{ (a_1,a_2,a_3,a_4,a_5) \in \mathbb{F}^5 \;|\; a_1-a_3-a_4=0 \}$
\item $W_2 = \{ (a_1,a_2,a_3,a_4,a_5) \in \mathbb{F}^5 \;|\; a_1=a_3+a_5 \text{ and } a_2-a_4=1 \}$ 
\item $W_3 = \{ (a_1,a_2,a_3,a_4,a_5) \in \mathbb{F}^5 \;|\; a_2=a_3=a_4 \text{ and } a_1+a_5=0 \}$
\end{enumerate}

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