# UC-Berkeley-eye-catch

• Torsion Submodule, Integral Domain, and Zero Divisors Let $R$ be a ring with $1$. An element of the $R$-module $M$ is called a torsion element if $rm=0$ for some nonzero element $r\in R$. The set of torsion elements is denoted $\Tor(M)=\{m \in M \mid rm=0 \text{ for some nonzero} r\in R\}.$ (a) Prove that if $R$ is an […]
• Matrix $XY-YX$ Never Be the Identity Matrix Let $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that $XY-YX=I.$   Hint. Suppose that such matrices exist and consider the trace of the matrix $XY-YX$. Recall that the trace of […]
• Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by $W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.$ (a) Which of the following vectors are in the subset […]
• If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set? Suppose that $A$ is a real $n\times n$ matrix. (a) Is it true that $A$ must commute with its transpose? (b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set. Is it true that the rows of $A$ must also form an orthonormal set? (University of […]
• Are Linear Transformations of Derivatives and Integrations Linearly Independent? Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation). Let $V$ be the vector space of all linear transformations from $W$ to $W$. The addition and the scalar multiplication of $V$ […]
• Every Group of Order 72 is Not a Simple Group Prove that every finite group of order $72$ is not a simple group. Definition. A group $G$ is said to be simple if the only normal subgroups of $G$ are the trivial group $\{e\}$ or $G$ itself. Hint. Let $G$ be a group of order $72$. Use the Sylow's theorem and determine […]
• 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 […]