# linear=algebra-eye-catch2

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• Transpose of a Matrix and Eigenvalues and Related Questions Let $A$ be an $n \times n$ real matrix. Prove the followings. (a) The matrix $AA^{\trans}$ is a symmetric matrix. (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal. (c) The matrix $AA^{\trans}$ is non-negative definite. (An $n\times n$ […]
• If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$. Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$. Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.   Hint. It follows from […]
• The Cyclotomic Field of 8-th Roots of Unity is $\Q(\zeta_8)=\Q(i, \sqrt{2})$ Let $\zeta_8$ be a primitive $8$-th root of unity. Prove that the cyclotomic field $\Q(\zeta_8)$ of the $8$-th root of unity is the field $\Q(i, \sqrt{2})$.   Proof. Recall that the extension degree of the cyclotomic field of $n$-th roots of unity is given by […]
• Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$ Let $V$ be a vector space over the field of real numbers $\R$. Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.   Proof. Since $V$ is an $n$-dimensional vector space, it has a basis $B=\{\mathbf{v}_1, \dots, […] • A Simple Abelian Group if and only if the Order is a Prime Number Let G be a group. (Do not assume that G is a finite group.) Prove that G is a simple abelian group if and only if the order of G is a prime number. Definition. A group G is called simple if G is a nontrivial group and the only normal subgroups of G is […] • 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 […] • Non-Abelian Group of Order pq and its Sylow Subgroups Let G be a non-abelian group of order pq, where p, q are prime numbers satisfying q \equiv 1 \pmod p. Prove that a q-Sylow subgroup of G is normal and the number of p-Sylow subgroups are q. Hint. Use Sylow's theorem. To review Sylow's theorem, check […] • Ascending Chain of Submodules and Union of its Submodules Let R be a ring with 1. Let M be an R-module. Consider an ascending chain \[N_1 \subset N_2 \subset \cdots$ of submodules of $M$. Prove that the union $\cup_{i=1}^{\infty} N_i$ is a submodule of $M$.   Proof. To simplify the notation, let us […]