Algebra Qual 2020-01

Problem 1

Let $A$ be a real $n\times n$ matrix and let $A^{\mathrm{\top }}$ denote its transpose.
(a) Prove that $(A\mathbf{v})\cdot \mathbf{w}=\mathbf{v}\cdot (A^{\mathrm{\top }}\mathbf{w})$ for all vectors $\mathbf{v},\mathbf{w}\in {\mathbf{R}}^n$. Hint: Recall that the dot product $\mathbf{u}\cdot \mathbf{v}$ equals the matrix product ${\mathbf{u}}^{\mathrm{\top }}\mathbf{v}$.
(b) Suppose now $A$ is also symmetric, i.e., that $A^{\mathrm{\top }}=A$. Also suppose $\mathbf{v}$ and $\mathbf{w}$ are eigenvectors of $A$ with different eigenvalues. Prove that $\mathbf{v}$ and $\mathbf{w}$ are orthogonal.

Problem 2

Let $M_4(\mathbf{R})$ denote the 16-dimensional real vector space of $4\times 4$ matrices with real entries, in which the vectors are represented as matrices. Let $T:M_4(\mathbf{R})\to M_4(\mathbf{R})$ be the linear transformation defined by $T(A)=A-A^{\mathrm{\top }}$.
(a) Determine the dimension of $\ker (T)$.
(b) Determine the dimension of $\mathrm{im}(T)$.

Problem 3

Let $G$ be a group. For each $a\in G$, let ${\gamma }_a$ denote the automorphism of $G$ defined by ${\gamma }_a(b)=ab{a}^{-1}$ for all $b\in G$. The set $\mathrm{Inn}(G)=\{{\gamma }_a:a\in G\}$ is a subgroup of the automorphism group of $G$, called the subgroup of inner automorphisms.

Prove that $\mathrm{Inn}(G)$ is isomorphic to $G/Z(G)$, where $Z(G)$ is the center of $G$.

Problem 4

Let ${\mathbf{Z}}_n$ denote the cyclic group of order $n$. Suppose $m\in \mathbf{N}$ is relatively prime to $n$. Define the function ${\mu }_m:{\mathbf{Z}}_n\to {\mathbf{Z}}_n$ by ${\mu }_m[a{]}_n=[ma{]}_n$.
(a) Prove that the map ${\mu }_m$ is a well-defined automorphism of ${\mathbf{Z}}_n$.
(b) Prove that any automorphism of ${\mathbf{Z}}_n$ has the form ${\mu }_m$ for some $m$.

Problem 5

Let $F$ be a field and $F[x]$ be the polynomial ring, which is a principal ideal domain. Let $R=\{f\in F[x]:f^{\prime }\in (x)\}$, where $(x)\subset F[x]$ is the ideal generated by $x$, and $f^{\prime }$ is the (formal) derivative of the polynomial $f$. It is a fact that $R$ is a subring of $F[x]$.
(a) Prove that $x^2$ and $x^3$ are irreducible elements of $R$.
(b) Let $(x^2,x^3)$ be the ideal in $R$ generated by $x^2$ and $x^3$. Prove this is a proper ideal of $R$.
(c) Prove that $(x^2,x^3)$ is not a principal ideal of $R$.