# MIT-exam-eye-catch

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- Eckmann–Hilton Argument: Group Operation is a Group Homomorphism Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying \[\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}\] for any $g\in G$. Let $\mu: G\times G \to G$ be a map defined […]
- Prove a Group is Abelian if $(ab)^2=a^2b^2$ Let $G$ be a group. Suppose that \[(ab)^2=a^2b^2\] for any elements $a, b$ in $G$. Prove that $G$ is an abelian group. Proof. To prove that $G$ is an abelian group, we need \[ab=ba\] for any elements $a, b$ in $G$. By the given […]
- Summary: Possibilities for the Solution Set of a System of Linear Equations In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems. Determine all possibilities for the solution set of the system of linear equations described below. (a) A homogeneous system of $3$ […]
- A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$ An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that $AB=I$, and $BA=I$, where $I$ is the $n\times n$ identity matrix. If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted […]
- The Union of Two Subspaces is Not a Subspace in a Vector Space Let $U$ and $V$ be subspaces of the vector space $\R^n$. If neither $U$ nor $V$ is a subset of the other, then prove that the union $U \cup V$ is not a subspace of $\R^n$. Proof. Since $U$ is not contained in $V$, there exists a vector $\mathbf{u}\in U$ but […]
- The Normalizer of a Proper Subgroup of a Nilpotent Group is Strictly Bigger Let $G$ be a nilpotent group and let $H$ be a proper subgroup of $G$. Then prove that $H \subsetneq N_G(H)$, where $N_G(H)$ is the normalizer of $H$ in $G$. Proof. Note that we always have $H \subset N_G(H)$. Hence our goal is to find an element in […]
- Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$ Let \[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\] be an ideal of the ring \[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\] Then determine the quotient ring $\Z[\sqrt{10}]/P$. Is $P$ a prime ideal? Is $P$ a maximal ideal? Solution. We […]
- How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}\] by finding a nonsingular […]