Algebra Qual 2018-09

Problem 1

Let $T : R^{3} \to R^{3}$ be the linear transformation defined by $T ([\begin{matrix} x \\ y \\ z \end{matrix}]) = [\begin{matrix} x + y \\ 2 z - x \\ y + 2 z \end{matrix}]$ .

Find the matrix that represents $T$ with respect to the standard basis for $R^{3}$ .
Find a basis for the kernel of $T$ .
Determine the rank of $T$ .

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Let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation defined by $T\left(\begin{bmatrix} x \\ y \\ z\end{bmatrix}\right) = \begin{bmatrix}  x+y \\ 2z-x \\ y+2z\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Find the matrix that represents $T$ with respect to the standard basis for ${\bf R}^3$.
	\item Find a basis for the kernel of $T$.
	\item Determine the rank of $T$.
\end{enumerate}

Problem 2

Suppose $G$ is a group, $H \leq G$ a subgroup, and $a, b \in G$ . Prove that the following are equivalent:

$a H = b H$
$b \in a H$
$b^{- 1} a \in H$

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Suppose $G$ is a group, $H\leq G$ a subgroup, and $a,b\in G$. Prove that the following are equivalent:
\begin{enumerate}[label=\alph*)]
	\item $aH=bH$
	\item $b\in aH$
	\item $b^{-1}a\in H$
\end{enumerate}

Problem 3

Let $G$ be a group and $H, K ⊴ G$ be normal subgroups with $H \cap K = {e}$ . Show that each element in $H$ commutes with every element in $K$ .

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Let $G$ be a group and $H,K\mathrel{\unlhd}G$ be normal subgroups with $H\cap K=\{e\}$. Show that each element in $H$ commutes with every element in $K$.

Problem 4

Let $R$ be a commutative ring with unity.

Define what it means for an element in $R$ to be prime, and also what it means for an element to be irreducible.
Prove that if $R$ is an integral domain, then every prime element is irreducible.

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Let $R$ be a commutative ring with unity.
\begin{enumerate}[label=\alph*)]
	\item Define what it means for an element in $R$ to be {\bfseries prime}, and also what it means for an element to be {\bfseries irreducible}.
	\item Prove that if $R$ is an integral domain, then every prime element is irreducible.
\end{enumerate}

Problem 5

Suppose $A$ is a real $n \times n$ matrix that satisfies $A^{2} v = 2 A v$ for every $v \in R^{n}$ .

Show that the only possible eigenvalues of $A$ are 0 and 2.
For each $λ \in R$ , let $E_{λ}$ denote the $λ$ -eigenspace of $A$ , i.e., $E_{λ} = {v \in R^{n} ∣ A v = λ v}$ . Prove that $R^{n} = E_{0} \oplus E_{2}$ . (Hint: For every vector $v$ one can write $v = (v - \frac{1}{2} A v) + \frac{1}{2} A v$ .)

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L A T E X

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Suppose $A$ is a real $n\times n$ matrix that satisfies $A^2 {\bf v} = 2A{\bf v}$ for every ${\bf v}\in {\bf R}^n$.
\begin{enumerate}[label=\alph*)]
	\item Show that the only possible eigenvalues of $A$ are 0 and 2.
	\item For each $\lambda\in {\bf R}$, let $E_{\lambda}$ denote the $\lambda$-eigenspace of $A$, i.e., $E_{\lambda} = \{{\bf v}\in {\bf R}^n\mid A{\bf v}=\lambda {\bf v}\}$. Prove that ${\bf R}^n = E_0\oplus E_2$. ({\itshape Hint:} For every vector ${\bf v}$ one can write ${\bf v}=({\bf v}-\frac{1}{2}A{\bf v})+\frac{1}{2}A{\bf v}$.)
\end{enumerate}