# mmc%20rep%20of%20mcg

mmc%20rep%20of%20mcg

• Find a Matrix that Maps Given Vectors to Given Vectors Suppose that a real matrix $A$ maps each of the following vectors $\mathbf{x}_1=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ into the […]
• Automorphism Group of $\Q(\sqrt[3]{2})$ Over $\Q$. Determine the automorphism group of $\Q(\sqrt[3]{2})$ over $\Q$. Proof. Let $\sigma \in \Aut(\Q(\sqrt[3]{2}/\Q)$ be an automorphism of $\Q(\sqrt[3]{2})$ over $\Q$. Then $\sigma$ is determined by the value $\sigma(\sqrt[3]{2})$ since any element $\alpha$ of $\Q(\sqrt[3]{2})$ […]
• Every Group of Order 20449 is an Abelian Group Prove that every group of order $20449$ is an abelian group.   Outline of the Proof Note that $20449=11^2 \cdot 13^2$. Let $G$ be a group of order $20449$. We prove by Sylow's theorem that there are a unique Sylow $11$-subgroup and a unique Sylow $13$-subgroup of […]
• If the Quotient Ring is a Field, then the Ideal is Maximal Let $R$ be a ring with unit $1\neq 0$. Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$. (Do not assume that the ring $R$ is commutative.)   Proof. Let $I$ be an ideal of $R$ such that $M \subset I \subset […] • 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 […] • Determine Whether Each Set is a Basis for \R^3 Determine whether each of the following sets is a basis for \R^3. (a) S=\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} […] • Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix Let V be the vector space of all 3\times 3 real matrices. Let A be the matrix given below and we define \[W=\{M\in V \mid AM=MA\}.$ That is, $W$ consists of matrices that commute with $A$. Then $W$ is a subspace of $V$. Determine which matrices are in the subspace $W$ […]
• Basis of Span in Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let $S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}$ be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]