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• Solve the System of Linear Equations and Give the Vector Form for the General Solution Solve the following system of linear equations and give the vector form for the general solution. \begin{align*} x_1 -x_3 -2x_5&=1 \\ x_2+3x_3-x_5 &=2 \\ 2x_1 -2x_3 +x_4 -3x_5 &= 0 \end{align*} (The Ohio State University, linear algebra midterm exam […]
• Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$ Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$. Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$. Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.   Proof. Define the map $\psi: R[x,y] \to […]
• Commutator Subgroup and Abelian Quotient Group Let $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$. Let $N$ be a subgroup of $G$. Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.   Definitions. Recall that for any $a, b \in G$, the […]
• If the Quotient by the Center is Cyclic, then the Group is Abelian Let $Z(G)$ be the center of a group $G$. Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian. Steps. Write $G/Z(G)=\langle \bar{g} \rangle$ for some $g \in G$. Any element $x\in G$ can be written as $x=g^a z$ for some $z \in Z(G)$ and $a \in \Z$. Using […]
• Orthogonal Nonzero Vectors Are Linearly Independent Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$. Suppose that $S$ is an orthogonal set. (a) Show that $S$ is linearly independent. (b) If $k=n$, then prove that $S$ is a basis for $\R^n$.   Proof. (a) […]
• There is Exactly One Ring Homomorphism From the Ring of Integers to Any Ring Let $\Z$ be the ring of integers and let $R$ be a ring with unity. Determine all the ring homomorphisms from $\Z$ to $R$.   Definition. Recall that if $A, B$ are rings with unity then a ring homomorphism $f: A \to B$ is a map […]
• Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable For which values of constants $a, b$ and $c$ is the matrix \[A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}\] diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the […]
• Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$ Let $D_8$ be the dihedral group of order $8$. Using the generators and relations, we have \[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.\] (a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$. Prove that the centralizer […]