Algebra Qual 2021-01

Suppose $\{{\bf v}_1,{\bf v}_2,{\bf v}_3\}$ is a basis for ${\bf R}^3$ and $T:{\bf R}^3\to {\bf R}^3$ is a linear transformation satisfying the following:
\begin{align*}
	T({\bf v}_1) &= 4{\bf v}_1+2{\bf v}_2\\
	T({\bf v}_2) &= 5{\bf v}_2\\
	T({\bf v}_3) &= -2{\bf v}_1+4{\bf v}_2+5{\bf v}_3.
\end{align*}
Determine the eigenvalues of $T$ and find a basis for each eigenspace.

\begin{enumerate}[label=\alph*)]
	\item Give an explicit example (with proof) showing that the union of two subspaces (of a given vector space) is not necessarily a subspace.
	\item Suppose $U_1$ and $U_2$ are subspaces of a vector space $V$. Recall that their {\bfseries sum} is defined to be the set $U_1+U_2 =\left\{u_1+u_2\,\mid \, u_1\in U_1, u_2\in U_2\right\}$. Prove $U_1+U_2$ is a subspace of $V$ containing $U_1$ and $U_2$.
\end{enumerate}

Let $H$ be a subgroup of a group $G$. The {\bfseries normalizer} of $H$ in $G$ is the set ${\bf N}_G(H)=\{g\in G\,\mid\, gH=Hg\}$.
\begin{enumerate}[label=\alph*)]
	\item Prove ${\bf N}_G(H)$ is a subgroup of $G$ containing $H$.
	\item Prove ${\bf N}_G(H)$ is the largest subgroup of $G$ in which $H$ is normal.
\end{enumerate}

Suppose $G$ is a group and $N\unlhd G$ is a finite normal subgroup. Prove that if $G/N$ contains an element of order $n$, then $G$ also contains an element of order $n$.

Let $R$ be a commutative ring with unity, let $I\subseteq R$ be an ideal, and let $\pi:R\to R/I$ be the natural projection homomorphism.
\begin{enumerate}[label=\alph*)]
	\item Show that if $\wp$ is a prime ideal of $R/I$, then $\pi^{-1}(\wp)$ is a prime ideal of $R$.
	\item Show that the assignment $\wp\mapsto\pi^{-1}(\wp)$ is injective on the set of prime ideals of $R/I$.
\end{enumerate}