Algebra Qual 2019-09

Problem 1

Let $P_3$ be the real vector space of all real polynomials of degree three or less. Define $L:P_3\to P_3$ by $L(p(x))=p(x)+p(-x)$.
(a) Prove $L$ is a linear transformation.
(b) Find a basis for the null space of $L$.
(c) Compute the dimension of the image of $L$.

Problem 2

Let $T:{\mathbf{R}}^3\to {\mathbf{R}}^3$ be the linear transformation that rotates counterclockwise around the $z$-axis by $\frac{2\pi }{3}$.
(a) Write the matrix for $T$ with respect to the standard basis $\{\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\}$.
(b) Write the matrix for $T$ with respect to the basis $\{\left[\begin{array}{c}\frac{\sqrt{3}}{2}\\ -\frac{1}{2}\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\}$.
(c) Determine all (complex) eigenvalues of $T$.
(d) Is $T$ diagonalizable over $\mathbf{C}$? Justify your answer.

Problem 3

Suppose $G$ is a cyclic group of finite order $n$, and $t\in G$ is a generator.
(a) Give a positive integer $d$ such that $t^{-1}=t^d$.
(b) Let $c$ be an integer and let $m=gcd(n,c)$. Prove that the order of $t^c$ is $\frac{n}{m}$.

Problem 4

Suppose $G$ is a group, $H$ and $K$ are normal subgroups of $G$, and $H\le K$.
(a) Define a group homomorphism from $K$ to $G/H$.
(b) Compute the kernel of the homomorphism in (a), and apply the First Isomorphism Theorem.

Problem 5

Let $I\subseteq \mathbf{Z}[x]$ denote the set of all polynomials with even constant term.
(a) Prove that $I$ is an ideal.
(b) Prove that $I$ is not a principal ideal.