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Kyoto University Exam problems and solutions in mathematrics


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  • The Product of Two Nonsingular Matrices is NonsingularThe Product of Two Nonsingular Matrices is Nonsingular Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix. (The Ohio State University, Linear Algebra Final Exam Problem)   Definition (Nonsingular Matrix) An $n\times n$ matrix is called nonsingular if the […]
  • Find a Formula for a Linear TransformationFind a Formula for a Linear Transformation If $L:\R^2 \to \R^3$ is a linear transformation such that \begin{align*} L\left( \begin{bmatrix} 1 \\ 0 \end{bmatrix}\right) =\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \,\,\,\, L\left( \begin{bmatrix} 1 \\ 1 \end{bmatrix}\right) =\begin{bmatrix} 2 \\ 3 […]
  • Algebraic Number is an Eigenvalue of Matrix with Rational EntriesAlgebraic Number is an Eigenvalue of Matrix with Rational Entries A complex number $z$ is called algebraic number (respectively, algebraic integer) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients. Prove that $z \in \C$ is an algebraic number (resp. algebraic integer) if and only if $z$ is an eigenvalue of […]
  • 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$ In the ring \[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\] show that $5$ is a prime element but $7$ is not a prime element.   Hint. An element $p$ in a ring $R$ is prime if $p$ is non zero, non unit element and whenever $p$ divide $ab$ for $a, b \in R$, then $p$ […]
  • A Group of Order the Square of a Prime is AbelianA Group of Order the Square of a Prime is Abelian Suppose the order of a group $G$ is $p^2$, where $p$ is a prime number. Show that (a) the group $G$ is an abelian group, and (b) the group $G$ is isomorphic to either $\Zmod{p^2}$ or $\Zmod{p} \times \Zmod{p}$ without using the fundamental theorem of abelian […]
  • The Order of a Conjugacy Class Divides the Order of the GroupThe Order of a Conjugacy Class Divides the Order of the Group Let $G$ be a finite group. The centralizer of an element $a$ of $G$ is defined to be \[C_G(a)=\{g\in G \mid ga=ag\}.\] A conjugacy class is a set of the form \[\Cl(a)=\{bab^{-1} \mid b\in G\}\] for some $a\in G$. (a) Prove that the centralizer of an element of $a$ […]
  • Prove that a Group of Order 217 is Cyclic and Find the Number of GeneratorsProve that a Group of Order 217 is Cyclic and Find the Number of Generators Let $G$ be a finite group of order $217$. (a) Prove that $G$ is a cyclic group. (b) Determine the number of generators of the group $G$.     Sylow's Theorem We will use Sylow's theorem to prove part (a). For a review of Sylow's theorem, check out the […]
  • Orthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct EigenvaluesOrthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct Eigenvalues Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$. Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$. (Nagoya University, Linear Algebra Final Exam Problem)   Hint. Two […]

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