Algebra Qual 2019-06

Let $L$ be the line in ${\bf R}^2$ defined by $y=-3x$, and let $T:{\bf R}^2\to {\bf R}^2$ be the linear transformation that orthogonally projects onto $L$ and then stretches along $L$ by a factor of two.
\begin{enumerate}[label=\alph*)]
	\item Find the eigenvalues and an eigenbasis $\mathcal{B}$ for $T$.
	\item Determine the matrix for $T$ with respect to the basis $\mathcal{B}$.
	\item Determine the matrix for $T$ with respect to the standard basis.
\end{enumerate}

Let $G$ be a group and $H\leq G$ a subgroup. For each coset $aH$ of $H$ in $G$, define the set
\[
	G_{aH}=\{b\in G\,|\,baH=aH\}.
\]
\begin{enumerate}[label=\alph*)]
	\item Prove that $G_{aH}$ is a subgroup of $G$.
	\item Suppose that $H$ is normal in $G$. Prove that $G_{aH}=H$.
\end{enumerate}

Suppose $G_1$ and $G_2$ are groups, with identity elements $e_1$ and $e_2$, respectively. Prove that if $\phi:G_1\to G_2$ is an isomorphism, then $\phi(e_1)=e_2$.

Let $R$ be a commutative ring. For each nonempty subset $X\subseteq R$, the {\bfseries annihilator} of $X$ is the set $\operatorname{ann}(X)=\{a\in R\mid ax=0\text{ for all }x\in X\}$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $\operatorname{ann}(X)$ is an ideal of $R$.
	\item Prove that $X\subseteq \operatorname{ann}(\operatorname{ann}(X))$.
\end{enumerate}

\begin{enumerate}[label=\alph*)]
	\item Prove that for every commutative ring with unity, $R$, there is a unique ring homomorphism $\phi_R: {\bf Z}\to R$, and that $\ker(\phi_R)=\langle d_R\rangle$ for some unique nonnegative integer $d_R$. The number $d_R$ is called the {\bfseries characteristic} of $R$ and is denoted $\operatorname{char}(R)$.
	\item Suppose $F_1$ and $F_2$ are fields for which there exists a ring homomorphism $f:F_1\to F_2$. Prove that $\operatorname{char}(F_1)=\operatorname{char}(F_2)$.
\end{enumerate}