Algebra Qual 2021-01

Problem 1

Suppose ${v_{1}, v_{2}, v_{3}}$ is a basis for $R^{3}$ and $T : R^{3} \to R^{3}$ is a linear transformation satisfying the following:

\begin{aligned} T (v_{1}) & = 4 v_{1} + 2 v_{2} \\ T (v_{2}) & = 5 v_{2} \\ T (v_{3}) & = - 2 v_{1} + 4 v_{2} + 5 v_{3} . \end{aligned}

Determine the eigenvalues of $T$ and find a basis for each eigenspace.

Problem 2

(a) Give an explicit example (with proof) showing that the union of two subspaces (of a given vector space) is not necessarily a subspace.
(b) Suppose $U_{1}$ and $U_{2}$ are subspaces of a vector space $V$ . Recall that their sum is defined to be the set $U_{1} + U_{2} = {u_{1} + u_{2} ∣ u_{1} \in U_{1}, u_{2} \in U_{2}}$ . Prove $U_{1} + U_{2}$ is a subspace of $V$ containing $U_{1}$ and $U_{2}$ .

Problem 3

Let $H$ be a subgroup of a group $G$ . The normalizer of $H$ in $G$ is the set $N_{G} (H) = {g \in G ∣ g H = H g}$ .
(a) Prove $N_{G} (H)$ is a subgroup of $G$ containing $H$ .
(b) Prove $N_{G} (H)$ is the largest subgroup of $G$ in which $H$ is normal.

Problem 4

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

Problem 5

Let $R$ be a commutative ring with unity, let $I \subseteq R$ be an ideal, and let $π : R \to R / I$ be the natural projection homomorphism.
(a) Show that if $℘$ is a prime ideal of $R / I$ , then $π^{- 1} (℘)$ is a prime ideal of $R$ .
(b) Show that the assignment $℘ \mapsto π^{- 1} (℘)$ is injective on the set of prime ideals of $R / I$ .