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  • Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)Trace, Determinant, and Eigenvalue (Harvard University Exam Problem) (a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$. Find $\det(A)$. (b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$. (c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of […]
  • Non-Abelian Group of Order $pq$ and its Sylow SubgroupsNon-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 […]
  • All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8 Determine all the conjugacy classes of the dihedral group \[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\] of order $8$. Hint. You may directly compute the conjugates of each element but we are going to use the following theorem to simplify the […]
  • Show the Subset of the Vector Space of Polynomials is a Subspace and Find its BasisShow the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient. Let $W$ be the following subset of $P_3$. \[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\] Here $p'(x)$ is the first derivative of $p(x)$ and […]
  • Exponential Functions Form a Basis of a Vector SpaceExponential Functions Form a Basis of a Vector Space Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let \[V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}\] be a subset in $C[-1, 1]$. (a) Prove that $V$ is a subspace of $C[-1, 1]$. (b) […]
  • Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$ Let $A$ be an $n\times n$ nonsingular matrix with integer entries. Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.   Hint. If $B$ is a square matrix whose entries are integers, then the […]
  • The Product Distinct Sylow $p$-Subgroups Can Never be a SubgroupThe Product 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 $K$ […]
  • Ascending Chain of Submodules and Union of its SubmodulesAscending 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 […]

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