Algebra Qual 2019-06

Problem 1

Let $L$ be the line in $R^{2}$ defined by $y = - 3 x$ , and let $T : R^{2} \to R^{2}$ be the linear transformation that orthogonally projects onto $L$ and then stretches along $L$ by a factor of two.
(a) Find the eigenvalues and an eigenbasis $B$ for $T$ .
(b) Determine the matrix for $T$ with respect to the basis $B$ .
(c) Determine the matrix for $T$ with respect to the standard basis.

Problem 2

Let $G$ be a group and $H \leq G$ a subgroup. For each coset $a H$ of $H$ in $G$ , define the set

G_{a H} = {b \in G | b a H = a H} .

(a) Prove that $G_{a H}$ is a subgroup of $G$ .
(b) Suppose that $H$ is normal in $G$ . Prove that $G_{a H} = H$ .

Problem 3

Suppose $G_{1}$ and $G_{2}$ are groups, with identity elements $e_{1}$ and $e_{2}$ , respectively. Prove that if $ϕ : G_{1} \to G_{2}$ is an isomorphism, then $ϕ (e_{1}) = e_{2}$ .

Problem 4

Let $R$ be a commutative ring. For each nonempty subset $X \subseteq R$ , the annihilator of $X$ is the set $ann (X) = {a \in R ∣ a x = 0 for all x \in X}$ .
(a) Prove that $ann (X)$ is an ideal of $R$ .
(b) Prove that $X \subseteq ann (ann (X))$ .

Problem 5

(a) Prove that for every commutative ring with unity, $R$ , there is a unique ring homomorphism $ϕ_{R} : Z \to R$ , and that $\ker (ϕ_{R}) = ⟨ d_{R} ⟩$ for some unique nonnegative integer $d_{R}$ . The number $d_{R}$ is called the characteristic of $R$ and is denoted $char (R)$ .
(b) Suppose $F_{1}$ and $F_{2}$ are fields for which there exists a ring homomorphism $f : F_{1} \to F_{2}$ . Prove that $char (F_{1}) = char (F_{2})$ .