# cropped-question-logo.jpg

• Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let $W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}$ be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.   Proof. […]
• The Product of Distinct Sylow $p$-Subgroups Can Never be a Subgroup Let $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$. Prove that the product $HK$ can never be a subgroup of the group $G$.   Hint. Use the following fact. If $H$ and $K$ […]
• Similar Matrices Have the Same Eigenvalues Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same. Proof. We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]
• Diagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix $A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}$ is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
• A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers. Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$. Hint. Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix. […]
• Submodule Consists of Elements Annihilated by Some Power of an Ideal Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$. Let $M'$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$. Prove that $M'$ is a submodule of […]
• Extension Degree of Maximal Real Subfield of Cyclotomic Field Let $n$ be an integer greater than $2$ and let $\zeta=e^{2\pi i/n}$ be a primitive $n$-th root of unity. Determine the degree of the extension of $\Q(\zeta)$ over $\Q(\zeta+\zeta^{-1})$. The subfield $\Q(\zeta+\zeta^{-1})$ is called maximal real subfield.   Proof. […]
• A Group Homomorphism and an Abelian Group Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$. Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ […]