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Nagoya University Linear Algebra Exam Problems and Solutions


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  • If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal SubgroupIf Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup Let $H$ be a subgroup of a group $G$. Suppose that for each element $x\in G$, we have $x^2\in H$. Then prove that $H$ is a normal subgroup of $G$. (Purdue University, Abstract Algebra Qualifying Exam)   Proof. To show that $H$ is a normal subgroup of […]
  • Find a Basis for the Subspace spanned by Five VectorsFind a Basis for the Subspace spanned by Five Vectors Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where \[ \mathbf{v}_{1}= \begin{bmatrix} 1 \\ 2 \\ 2 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{2}= \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \end{bmatrix} ,\;\mathbf{v}_{3}= \begin{bmatrix} 1 \\ 5 \\ -1 […]
  • Normalizer and Centralizer of a Subgroup of Order 2Normalizer and Centralizer of a Subgroup of Order 2 Let $H$ be a subgroup of order $2$. Let $N_G(H)$ be the normalizer of $H$ in $G$ and $C_G(H)$ be the centralizer of $H$ in $G$. (a) Show that $N_G(H)=C_G(H)$. (b) If $H$ is a normal subgroup of $G$, then show that $H$ is a subgroup of the center $Z(G)$ of […]
  • Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$ Let $A$ be the coefficient matrix of the system of linear equations \begin{align*} -x_1-2x_2&=1\\ 2x_1+3x_2&=-1. \end{align*} (a) Solve the system by finding the inverse matrix $A^{-1}$. (b) Let $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ be the solution […]
  • Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ Let \[A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}.\] Find $A^{10}\mathbf{v}$. You may use the following information without proving […]
  • If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal SubgroupIf 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 […]
  • Every Cyclic Group is AbelianEvery Cyclic Group is Abelian Prove that every cyclic group is abelian.   Proof. Let $G$ be a cyclic group with a generator $g\in G$. Namely, we have $G=\langle g \rangle$ (every element in $G$ is some power of $g$.) Let $a$ and $b$ be arbitrary elements in $G$. Then there exists […]
  • Find an Orthonormal Basis of the Given Two Dimensional Vector SpaceFind an Orthonormal Basis of the Given Two Dimensional Vector Space Let $W$ be a subspace of $\R^4$ with a basis \[\left\{\, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} \,\right\}.\] Find an orthonormal basis of $W$. (The Ohio State […]

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