# univeristy-of-tokyo-eye-catch

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• How Many Solutions for $x+x=1$ in a Ring? Is there a (not necessarily commutative) ring $R$ with $1$ such that the equation $x+x=1$ has more than one solutions $x\in R$?   Solution. We claim that there is at most one solution $x$ in the ring $R$. Suppose that we have two solutions $r, s \in R$. That is, we […]
• Isomorphism Criterion of Semidirect Product of Groups Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism. The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation $(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),$ where $a_i […] • If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field. Let$R$be a commutative ring with$1$. Prove that if every proper ideal of$R$is a prime ideal, then$R$is a field. Proof. As the zero ideal$(0)$of$R$is a proper ideal, it is a prime ideal by assumption. Hence$R=R/\{0\}$is an integral […] • The Product of Two Nonsingular Matrices is Nonsingular Prove that if$n\times n$matrices$A$and$B$are nonsingular, then the product$AB$is also a nonsingular matrix. (The Ohio State University, Linear Algebra Final Exam Problem) Definition (Nonsingular Matrix) An$n\times n$matrix is called nonsingular if the […] • Prove that the Center of Matrices is a Subspace Let$V$be the vector space of$n \times n$matrices with real coefficients, and define $W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.$ The set$W$is called the center of$V$. Prove that$W$is a subspace […] • Determinant/Trace and Eigenvalues of a Matrix Let$A$be an$n\times n$matrix and let$\lambda_1, \dots, \lambda_n$be its eigenvalues. Show that (1) $$\det(A)=\prod_{i=1}^n \lambda_i$$ (2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$ Here$\det(A)$is the determinant of the matrix$A$and$\tr(A)$is the trace of the matrix […] • Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors Let$V$be a vector space and$B$be a basis for$V$. Let$\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$be vectors in$V$. Suppose that$A$is the matrix whose columns are the coordinate vectors of$\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […]
• A Group of Linear Functions Define the functions $f_{a,b}(x)=ax+b$, where $a, b \in \R$ and $a>0$. Show that $G:=\{ f_{a,b} \mid a, b \in \R, a>0\}$ is a group . The group operation is function composition. Steps. Check one by one the followings. The group operation on $G$ is […]