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Tokyo University Linear Algebra Exam Problems and Solutions


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  • If a Subgroup Contains a Sylow Subgroup, then the Normalizer is the Subgroup itselfIf a Subgroup Contains a Sylow Subgroup, then the Normalizer is the Subgroup itself Let $G$ be a finite group and $P$ be a nontrivial Sylow subgroup of $G$. Let $H$ be a subgroup of $G$ containing the normalizer $N_G(P)$ of $P$ in $G$. Then show that $N_G(H)=H$.   Hint. Use the conjugate part of the Sylow theorem. See the second statement of the […]
  • Simple Commutative Relation on MatricesSimple Commutative Relation on Matrices Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that \[A(A+B)^{-1}B=B(A+B)^{-1}A.\] (University of California, Berkeley Qualifying Exam) Proof. Let $P=A+B$. Then $B=P-A$. Using these, we express the given […]
  • Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$ Let $A$ be an $n \times n$ matrix and let $c$ be a complex number. (a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]
  • Exponential Functions are Linearly IndependentExponential Functions are Linearly Independent Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers. Show that exponential functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\] are linearly independent over $\R$. Hint. Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\] […]
  • Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a RelationQuiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation (a) Find the inverse matrix of \[A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason. (b) Find a nonsingular $2\times 2$ matrix $A$ such that \[A^3=A^2B-3A^2,\] where […]
  • Coordinate Vectors and Dimension of Subspaces (Span)Coordinate Vectors and Dimension of Subspaces (Span) Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$. Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of […]
  • A Group Homomorphism that Factors though Another GroupA Group Homomorphism that Factors though Another Group Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$. Define a map $\tilde{f}:H\to K$ as follows. For each $h\in H$, […]
  • If $R$ is a Noetherian Ring and $f:R\to R’$ is a Surjective Homomorphism, then $R’$ is NoetherianIf $R$ is a Noetherian Ring and $f:R\to R’$ is a Surjective Homomorphism, then $R’$ is Noetherian Suppose that $f:R\to R'$ is a surjective ring homomorphism. Prove that if $R$ is a Noetherian ring, then so is $R'$.   Definition. A ring $S$ is Noetherian if for every ascending chain of ideals of $S$ \[I_1 \subset I_2 \subset \cdots \subset I_k \subset […]

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