Kyushu-University-Linear-Algebra

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Kyushu University Linear Algebra Exam Problems and Solutions


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  • Even Perfect Numbers and Mersenne Prime NumbersEven Perfect Numbers and Mersenne Prime Numbers Prove that if $2^n-1$ is a Mersenne prime number, then \[N=2^{n-1}(2^n-1)\] is a perfect number. On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.   Definitions. In this post, a […]
  • A Simple Abelian Group if and only if the Order is a Prime NumberA 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 […]
  • Any Finite Group Has a Composition SeriesAny Finite Group Has a Composition Series Let $G$ be a finite group. Then show that $G$ has a composition series.   Proof. We prove the statement by induction on the order $|G|=n$ of the finite group. When $n=1$, this is trivial. Suppose that any finite group of order less than $n$ has a composition […]
  • A Homomorphism from the Additive Group of Integers to ItselfA Homomorphism from the Additive Group of Integers to Itself Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism. Then show that there exists an integer $a$ such that \[f(n)=an\] for any integer $n$.   Hint. Let us first recall the definition of a group homomorphism. A group homomorphism from a […]
  • The Center of a p-Group is Not TrivialThe Center of a p-Group is Not Trivial Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$. (Such a group is called a $p$-group.) Show that the center $Z(G)$ of the group $G$ is not trivial. Hint. Use the class equation. Proof. If $G=Z(G)$, then the statement is true. So suppose that $G\neq […]
  • The Sum of Cosine Squared in an Inner Product SpaceThe Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that \[\cos […]
  • Linearly Independent vectors $\mathbf{v}_1, \mathbf{v}_2$ and Linearly Independent Vectors $A\mathbf{v}_1, A\mathbf{v}_2$ for a Nonsingular MatrixLinearly Independent vectors $\mathbf{v}_1, \mathbf{v}_2$ and Linearly Independent Vectors $A\mathbf{v}_1, A\mathbf{v}_2$ for a Nonsingular Matrix Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix. (a) Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent. (b) If $\mathbf{v}_1, […]
  • Two Normal Subgroups Intersecting Trivially Commute Each OtherTwo Normal Subgroups Intersecting Trivially Commute Each Other Let $G$ be a group. Assume that $H$ and $K$ are both normal subgroups of $G$ and $H \cap K=1$. Then for any elements $h \in H$ and $k\in K$, show that $hk=kh$.   Proof. It suffices to show that $h^{-1}k^{-1}hk \in H \cap K$. In fact, if this it true then we have […]

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