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by Yu · Published · Updated

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- The Powers of the Matrix with Cosine and Sine Functions Prove the following identity for any positive integer $n$. \[\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos […]
- 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 […]
- Linear Transformation that Maps Each Vector to Its Reflection with Respect to $x$-Axis Let $F:\R^2\to \R^2$ be the function that maps each vector in $\R^2$ to its reflection with respect to $x$-axis. Determine the formula for the function $F$ and prove that $F$ is a linear transformation. Solution 1. Let $\begin{bmatrix} x \\ y […]
- If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain. Definitions: zero divisor, integral domain An element $a$ of a commutative ring $R$ is called a zero divisor […]
- Eigenvalues of a Matrix and its Transpose are the Same Let $A$ be a square matrix. Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$. Proof. Recall that the eigenvalues of a matrix are roots of its characteristic polynomial. Hence if the matrices $A$ and $A^{\trans}$ […]
- Eigenvalues of Orthogonal Matrices Have Length 1. Every $3\times 3$ Orthogonal Matrix Has 1 as an Eigenvalue (a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$. (b) Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an […]
- Condition that a Matrix is Similar to the Companion Matrix of its Characteristic Polynomial Let $A$ be an $n\times n$ complex matrix. Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Let $C$ be the companion matrix of the polynomial $p(x)$ given […]
- Generators of the Augmentation Ideal in a Group Ring Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by \[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\] where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring […]