# Yu-Tsumura-small

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• Complex 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 $\R^3$ 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 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’$is Noetherian Suppose that$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 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 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 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$x^p-2$. 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\$ […]