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  • Finitely Generated Torsion Module Over an Integral Domain Has a Nonzero AnnihilatorFinitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator (a) Let $R$ be an integral domain and let $M$ be a finitely generated torsion $R$-module. Prove that the module $M$ has a nonzero annihilator. In other words, show that there is a nonzero element $r\in R$ such that $rm=0$ for all $m\in M$. Here $r$ does not depend on […]
  • Abelian Normal Subgroup, Intersection, and Product of GroupsAbelian Normal Subgroup, Intersection, and Product of Groups Let $G$ be a group and let $A$ be an abelian subgroup of $G$ with $A \triangleleft G$. (That is, $A$ is a normal subgroup of $G$.) If $B$ is any subgroup of $G$, then show that \[A \cap B \triangleleft AB.\]   Proof. First of all, since $A \triangleleft G$, the […]
  • Sequences Satisfying Linear Recurrence Relation Form a SubspaceSequences Satisfying Linear Recurrence Relation Form a Subspace Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\] Let $U$ be the subset of $V$ defined by \[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\] Prove that $U$ is a subspace of […]
  • If the Matrix Product $AB=0$, then is $BA=0$ as Well?If the Matrix Product $AB=0$, then is $BA=0$ as Well? Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix. Is it true that the matrix product with opposite order $BA$ is also the zero matrix? If so, give a proof. If not, give a […]
  • Finite Integral Domain is a FieldFinite Integral Domain is a Field Show that any finite integral domain $R$ is a field.   Definition. A commutative ring $R$ with $1\neq 0$ is called an integral domain if it has no zero divisors. That is, if $ab=0$ for $a, b \in R$, then either $a=0$ or $b=0$. Proof. We give two proofs. Proof […]
  • Similar Matrices Have the Same EigenvaluesSimilar Matrices Have the Same Eigenvalues Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same. Proof. We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]
  • Linear Dependent/Independent Vectors of PolynomialsLinear Dependent/Independent Vectors of Polynomials Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent? (a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ […]
  • Inner Products, Lengths, and Distances of 3-Dimensional Real VectorsInner Products, Lengths, and Distances of 3-Dimensional Real Vectors For this problem, use the real vectors \[ \mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} . \] Suppose that $\mathbf{v}_4$ is another vector which is […]

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