Algebra Qual 2017-09

Problem 1

The additive group $Z = (Z, +)$ of rational integers is a subgroup of the additive group $Q = (Q, +)$ . Show that $Q$ has infinite index in $Q$ .

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The additive group ${\bf Z}=({\bf Z},+)$ of rational integers is a subgroup of the additive group ${\bf Q}=({\bf Q},+)$. Show that ${\bf Z}$ has infinite index in ${\bf Q}$.

Problem 2

Suppose $G$ is a finite group of even order.

Prove that an element in $G$ has order dividing 2 if and only if it is its own inverse.
Prove that the number of elements in $G$ of order 2 is odd.
Use (2) to show $G$ must contain a subgroup of order 2.

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Suppose $G$ is a finite group of even order.
\begin{enumerate}[label=\alph*)]
	\item Prove that an element in $G$ has order dividing 2 if and only if it is its own inverse.
	\item Prove that the number of elements in $G$ of order 2 is odd.
	\item Use (b) to show $G$ must contain a subgroup of order 2.
\end{enumerate}

Problem 3

Prove that every Euclidean domain is a principal ideal domain.

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Prove that every Euclidean domain is a principal ideal domain.

Problem 4

Let $L$ be the line $L$ parameterized by $L (t) = (2 t, - 3 t, t)$ for $t \in R$ , and let $T : R^{3} \to R^{3}$ be the linear transformation that is orthogonal projection onto $L$ .

Describe $\ker (T)$ and $im (T)$ , either implicitly (using equations in $x, y, z$ ) or parametrically.
List the eigenvalues of $T$ and their geometric multiplicities.
Find a basis for each eigenspace of $T$ .
Let $A$ be the matrix for $T$ with respect to the standard basis. Find a diagonal matrix $B$ and an invertible matrix $S$ such that $B = S^{- 1} A S$ . (You do not have to compute $A$ .)

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Let $L$ be the line $L$ parameterized by $L(t)=(2t,-3t,t)$ for $t\in {\bf R}$, and let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation that is orthogonal projection onto $L$.
\begin{enumerate}[label=\alph*)]
	\item Describe $\operatorname{ker}(T)$ and $\operatorname{im}(T)$, either implicitly (using equations in $x,y,z$) or parametrically.
	\item List the eigenvalues of $T$ and their geometric multiplicities.
	\item Find a basis for each eigenspace of $T$.
	\item Let $A$ be the matrix for $T$ with respect to the standard basis. Find a diagonal matrix $B$ and an invertible matrix $S$ such that $B=S^{-1}AS$. (You do not have to compute $A$.)
\end{enumerate}

Problem 5

Suppose $A$ is a $5 \times 5$ matrix and $v_{1}, v_{2}, v_{3}$ are eigenvectors of $A$ with distinct eigenvalues. Prove ${v_{1}, v_{2}, v_{3}}$ is a linearly independent set. Hint: Consider a minimal linear dependence relation.

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Suppose $A$ is a $5\times 5$ matrix and $v_1,v_2, v_3$ are eigenvectors of $A$ with distinct eigenvalues. Prove $\{v_1,v_2,v_3\}$ is a linearly independent set. {\itshape Hint:} Consider a minimal linear dependence relation.