Algebra Qual 2018-09

Problem 1

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

Problem 2

Suppose $G$ is a group, $H\le G$ a subgroup, and $a,b\in G$. Prove that the following are equivalent:
(a) $aH=bH$
(b) $b\in aH$
(c) $b^{-1}a\in H$

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 $K$.

Problem 4

Let $R$ be a commutative ring with unity.
(a) Define what it means for an element in $R$ to be prime, and also what it means for an element to be irreducible.
(b) Prove that if $R$ is an integral domain, then every prime element is irreducible.

Problem 5

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