# Nagoya-university-eye-catch

by Yu · Published · Updated

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- Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula \[T\left(\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \,\right)=\begin{bmatrix} x_1+3x_2-2x_3 \\ 2x_1+3x_2 \\ x_2+x_3 \end{bmatrix}.\] Determine whether $T$ is an […]
- The Ideal $(x)$ is Prime in the Polynomial Ring $R[x]$ if and only if the Ring $R$ is an Integral Domain Let $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain. Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a […]
- 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 […]
- If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$. Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix. Steps. Use the Jordan canonical form of the matrix $A$. We want […]
- Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible. Let \[A=\begin{bmatrix} 1 & 3 & 3 \\ -3 &-5 &-3 \\ 3 & 3 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 4 & 3 \\ -4 &-6 &-3 \\ 3 & 3 & 1 \end{bmatrix}.\] For this problem, you may use the fact that both matrices have the same characteristic […]
- Determine the Values of $a$ so that $W_a$ is a Subspace For what real values of $a$ is the set \[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\] a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions? Solution. The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]
- Linear Algebra Midterm 1 at the Ohio State University (1/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 1 and contains the […]
- 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 […]