Algebra Qual 2019-01

Problem 1

Let $W \subset R^{5}$ be the subspace spanned by the set of vectors ${⟨ 1, - 2, 0, 2, - 1 ⟩, ⟨ - 2, 4, - 1, 1, 2 ⟩, ⟨ 0, 1, 2, - 2, 1 ⟩}$ .

Compute the dimension of $W$ .
Determine the dimension of $W^{⊥}$ , the perpendicular subspace in $R^{5}$ .
Find a basis for $W^{⊥}$ .

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Let $W\subset {\bf R}^5$ be the subspace spanned by the set of vectors $\{\langle 1,-2,0,2,-1\rangle,\langle -2,4,-1,1,2\rangle,\langle 0,1,2,-2,1\rangle\}$.
\begin{enumerate}[label=(\alph*)]
	\item Compute the dimension of $W$.
	\item Determine the dimension of $W^\perp$, the perpendicular subspace in ${\bf R}^5$.
	\item Find a basis for $W^\perp$.
\end{enumerate}

Problem 2

Let $D$ be a principal ideal domain. Prove that every proper nonzero prime ideal is maximal.

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Let $D$ be a principal ideal domain. Prove that every proper nonzero prime ideal is maximal.

Problem 3

Let $G$ be a group and suppose $Aut (G)$ is trivial.

Show that $G$ is abelian.
Show that for any abelian group $H$ , the inversion map $ϕ (h) = h^{- 1}$ is an automorphism.
Use parts (1) and (2) above to show that $g^{2}$ is the identity element for every $g \in G$ .

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Let $G$ be a group and suppose $\operatorname{Aut}(G)$ is trivial.
\begin{enumerate}[label=(\alph*)]
	\item Show that $G$ is abelian.
	\item Show that for any abelian group $H$, the {\bfseries inversion map} $\phi(h)=h^{-1}$ is an automorphism.
	\item Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.
\end{enumerate}

Problem 4

Suppose $H$ is a group of order 15. Prove there does not exist a nontrivial group homomorphism $ϕ : D_{5} \to H$ , where $D_{5}$ is the dihedral group with ten elements.

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Suppose $H$ is a group of order 15. Prove there does not exist a nontrivial group homomorphism $\phi:D_5\to H$, where $D_5$ is the dihedral group with ten elements.

Problem 5

Let $a, b \in R$ and $T : R^{3} \to R^{3}$ be the linear transformation that is orthogonal projection onto the plane $z = a x + b y$ (with respect to the usual Euclidean inner-product on $R^{3}$ ).

Find the eigenvalues of $T$ and bases for the corresponding eigenspaces.
Is $T$ diagonalizable? Justify.
What is the characteristic polynomial of $T$ ?

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Let $a, b \in {\bf R}$ and $T: {\bf R}^3 \to {\bf R}^3$ be the linear transformation that is orthogonal projection onto the plane $z=ax+by$ (with respect to the usual Euclidean inner-product on ${\bf R}^3$).
\begin{enumerate}[label=(\alph*)]
	\item Find the eigenvalues of $T$ and bases for the corresponding eigenspaces.
	\item Is $T$ diagonalizable? Justify.
	\item What is the characteristic polynomial of $T$?
\end{enumerate}