# abelian-group-eye-catch

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• Elements of Finite Order of an Abelian Group form a Subgroup Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is, $H=\{ a\in G \mid \text{the order of a is finite}\}.$ Prove that $H$ is a subgroup of $G$.   Proof. Note that the identity element $e$ of […]
• Find Values of $h$ so that the Given Vectors are Linearly Independent Find the value(s) of $h$ for which the following set of vectors $\left \{ \mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} h \\ 1 \\ -h \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2h \\ 3h+1 […] • Is the Derivative Linear Transformation Diagonalizable? Let \mathrm{P}_2 denote the vector space of polynomials of degree 2 or less, and let T : \mathrm{P}_2 \rightarrow \mathrm{P}_2 be the derivative linear transformation, defined by \[ T( ax^2 + bx + c ) = 2ax + b .$ Is $T$ diagonalizable? If so, find a diagonal matrix which […]
• Determinant of Matrix whose Diagonal Entries are 6 and 2 Elsewhere Find the determinant of the following matrix $A=\begin{bmatrix} 6 & 2 & 2 & 2 &2 \\ 2 & 6 & 2 & 2 & 2 \\ 2 & 2 & 6 & 2 & 2 \\ 2 & 2 & 2 & 6 & 2 \\ 2 & 2 & 2 & 2 & 6 \end{bmatrix}.$ (Harvard University, Linear Algebra Exam […]
• Linear Algebra Midterm 1 at the Ohio State University (3/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 3 and contains […]
• Subset of Vectors Perpendicular to Two Vectors is a Subspace Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by $W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.$ Prove that the subset $W$ is a subspace of […]
• Injective Group Homomorphism that does not have Inverse Homomorphism Let $A=B=\Z$ be the additive group of integers. Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$. (a) Prove that $\phi$ is a group homomorphism. (b) Prove that $\phi$ is injective. (c) Prove that there does not exist a group homomorphism $\psi:B […] • Are Groups of Order 100, 200 Simple? Determine whether a group$G$of the following order is simple or not. (a)$|G|=100$. (b)$|G|=200$. Hint. Use Sylow's theorem and determine the number of$5$-Sylow subgroup of the group$G\$. Check out the post Sylow’s Theorem (summary) for a review of Sylow's […]