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  • Complex Conjugates of Eigenvalues of a Real Matrix are EigenvaluesComplex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues Let $A$ be an $n\times n$ real matrix. Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.   We give two proofs. Proof 1. Let $\mathbf{x}$ be an eigenvector corresponding to the […]
  • Linear Transformation and a Basis of the Vector Space $\R^3$Linear Transformation and a Basis of the Vector Space $\R^3$ Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$. Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$. Show […]
  • Basic Properties of Characteristic GroupsBasic Properties of Characteristic Groups Definition (automorphism). An isomorphism from a group $G$ to itself is called an automorphism of $G$. The set of all automorphism is denoted by $\Aut(G)$. Definition (characteristic subgroup). A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi […]
  • If $R$ is a Noetherian Ring and $f:R\to R’$ is a Surjective Homomorphism, then $R’$ is NoetherianIf $R$ is a Noetherian Ring and $f:R\to R’$ is a Surjective Homomorphism, then $R’$ is Noetherian Suppose that $f:R\to R'$ is a surjective ring homomorphism. Prove that if $R$ is a Noetherian ring, then so is $R'$.   Definition. A ring $S$ is Noetherian if for every ascending chain of ideals of $S$ \[I_1 \subset I_2 \subset \cdots \subset I_k \subset […]
  • How to Prove a Matrix is Nonsingular in 10 SecondsHow to Prove a Matrix is Nonsingular in 10 Seconds Using the numbers appearing in \[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\] we construct the matrix \[A=\begin{bmatrix} 3 & 14 &1592& 65358\\ 97932& 38462643& 38& 32\\ 7950& 2& 8841& 9716\\ 939937510& 5820& 974& […]
  • Every Finite Group Having More than Two Elements Has a Nontrivial AutomorphismEvery Finite Group Having More than Two Elements Has a Nontrivial Automorphism Prove that every finite group having more than two elements has a nontrivial automorphism. (Michigan State University, Abstract Algebra Qualifying Exam)   Proof. Let $G$ be a finite group and $|G|> 2$. Case When $G$ is a Non-Abelian Group Let us first […]
  • Every Plane Through the Origin in the Three Dimensional Space is a SubspaceEvery Plane Through the Origin in the Three Dimensional Space is a Subspace Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$.   Proof. Each plane $P$ in $\R^3$ through the origin is given by the equation \[ax+by+cz=0\] for some real numbers $a, b, c$. That is, the […]
  • Galois Group of the Polynomial  $x^p-2$.Galois Group of the Polynomial $x^p-2$. Let $p \in \Z$ be a prime number. Then describe the elements of the Galois group of the polynomial $x^p-2$.   Solution. The roots of the polynomial $x^p-2$ are \[ \sqrt[p]{2}\zeta^k, k=0,1, \dots, p-1\] where $\sqrt[p]{2}$ is a real $p$-th root of $2$ and $\zeta$ […]

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