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Kyoto University Exam problems and solutions in mathematrics


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  • Infinite Cyclic Groups Do Not Have Composition SeriesInfinite Cyclic Groups Do Not Have Composition Series Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.   Proof. Let $G=\langle a \rangle$ and suppose that $G$ has a composition series \[G=G_0\rhd G_1 \rhd \cdots G_{m-1} \rhd G_m=\{e\},\] where $e$ is the identity element of […]
  • The set of $2\times 2$ Symmetric Matrices is a SubspaceThe set of $2\times 2$ Symmetric Matrices is a Subspace Let $V$ be the vector space over $\R$ of all real $2\times 2$ matrices. Let $W$ be the subset of $V$ consisting of all symmetric matrices. (a) Prove that $W$ is a subspace of $V$. (b) Find a basis of $W$. (c) Determine the dimension of $W$.   Proof. […]
  • Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$ (a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where \[B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}.\] (b) Find the inverse matrix of $A$.   Solution (a) Find a $3\times 3$ nonsingular matrix $A$. Assume […]
  • Equivalent Conditions For a Prime Ideal in a Commutative RingEquivalent Conditions For a Prime Ideal in a Commutative Ring Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent: (a) The ideal $P$ is a prime ideal. (b) For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.   Proof. […]
  • Use Cramer’s Rule to Solve a $2\times 2$ System of Linear EquationsUse Cramer’s Rule to Solve a $2\times 2$ System of Linear Equations Use Cramer's rule to solve the system of linear equations \begin{align*} 3x_1-2x_2&=5\\ 7x_1+4x_2&=-1. \end{align*}   Solution. Let \[A=[A_1, A_2]=\begin{bmatrix} 3 & -2\\ 7& 4 \end{bmatrix},\] be the coefficient matrix of the system, where $A_1, A_2$ […]
  • Galois Group of the Polynomial  $x^p-2$.Galois Group of the Polynomial $x^p-2$. Let $p \in \Z$ be a prime number. Then describe the elements of the Galois group of the polynomial $x^p-2$.   Solution. The roots of the polynomial $x^p-2$ are \[ \sqrt[p]{2}\zeta^k, k=0,1, \dots, p-1\] where $\sqrt[p]{2}$ is a real $p$-th root of $2$ and $\zeta$ […]
  • Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$ Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$. Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.   Solution. Let $\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}$. Then we want \[\mathbf{v} […]
  • Subspaces of Symmetric, Skew-Symmetric MatricesSubspaces of Symmetric, Skew-Symmetric Matrices Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$. (a) The set $S$ consisting of all $n\times n$ symmetric matrices. (b) The set $T$ consisting of […]

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