Algebra Qual 2017-09

Problem 1

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

Problem 2

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

Problem 3

Prove that every Euclidean domain is a principal ideal domain.

Problem 4

Let $L$ be the line $L$ parameterized by $L(t)=(2t,-3t,t)$ for $t\in \mathbf{R}$, and let $T:{\mathbf{R}}^3\to {\mathbf{R}}^3$ be the linear transformation that is orthogonal projection onto $L$.
(a) Describe $\ker (T)$ and $\mathrm{im}(T)$, either implicitly (using equations in $x,y,z$) or parametrically.
(b) List the eigenvalues of $T$ and their geometric multiplicities.
(c) Find a basis for each eigenspace of $T$.
(d) 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 $A$.)

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.