# cody-slides

by Yu ·

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### More from my site

- A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer. Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$. Then prove that $H$ is a normal subgroup of $G$. (Michigan State University, Abstract Algebra Qualifying […]
- Idempotent Matrix and its Eigenvalues Let $A$ be an $n \times n$ matrix. We say that $A$ is idempotent if $A^2=A$. (a) Find a nonzero, nonidentity idempotent matrix. (b) Show that eigenvalues of an idempotent matrix $A$ is either $0$ or $1$. (The Ohio State University, Linear Algebra Final Exam […]
- Determine a Value of Linear Transformation From $\R^3$ to $\R^2$ Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that \[ T\left(\, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\,\right) =\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{ and }T\left(\, \begin{bmatrix} 0 \\ 1 \\ 1 […]
- The Matrix Exponential of a Diagonal Matrix For a square matrix $M$, its matrix exponential is defined by \[e^M = \sum_{i=0}^\infty \frac{M^k}{k!}.\] Suppose that $M$ is a diagonal matrix \[ M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & […]
- The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements Let $G$ be an abelian group. Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively. Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$. Also determine whether the statement is true if $G$ is a […]
- Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$ Let $A$ be an $n \times n$ matrix and let $c$ be a complex number. (a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]
- If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$ Let $N$ be a normal subgroup of a group $G$. Suppose that $G/N$ is an infinite cyclic group. Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$. Hint. Use the fourth (or Lattice) isomorphism theorem. Proof. […]
- The Centralizer of a Matrix is a Subspace Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define \[W = \{ A \in V \mid AM = MA \}.\] The set $W$ here is called the centralizer of $M$ in $V$. Prove that $W$ is a subspace of $V$. Proof. First we check that the zero […]