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  • In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal IdealIn a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$. Show that $P$ is a maximal ideal in $R$.   Definition A commutative ring $R$ is a principal ideal domain (PID) if $R$ is a domain and any ideal $I$ is generated by a single element […]
  • Any Subgroup of Index 2 in a Finite Group is NormalAny Subgroup of Index 2 in a Finite Group is Normal Show that any subgroup of index $2$ in a group is a normal subgroup. Hint. Left (right) cosets partition the group into disjoint sets. Consider both left and right cosets. Proof. Let $H$ be a subgroup of index $2$ in a group $G$. Let $e \in G$ be the identity […]
  • Order of the Product of Two Elements in an Abelian GroupOrder of the Product of Two Elements in an Abelian Group Let $G$ be an abelian group with the identity element $1$. Let $a, b$ be elements of $G$ with order $m$ and $n$, respectively. If $m$ and $n$ are relatively prime, then show that the order of the element $ab$ is $mn$.   Proof. Let $r$ be the order of the element […]
  • A Matrix Having One Positive Eigenvalue and One Negative EigenvalueA Matrix Having One Positive Eigenvalue and One Negative Eigenvalue Prove that the matrix \[A=\begin{bmatrix} 1 & 1.00001 & 1 \\ 1.00001 &1 &1.00001 \\ 1 & 1.00001 & 1 \end{bmatrix}\] has one positive eigenvalue and one negative eigenvalue. (University of California, Berkeley Qualifying Exam Problem)   Solution. Let us put […]
  • How to Find a Basis for the Nullspace, Row Space, and Range of a MatrixHow to Find a Basis for the Nullspace, Row Space, and Range of a Matrix Let $A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$. (a) Find a basis for the nullspace of $A$. (b) Find a basis for the row space of $A$. (c) Find a basis for the range of $A$ that consists of column vectors of $A$. (d) […]
  • If Every Nonidentity Element of a Group has Order 2, then it’s an Abelian GroupIf Every Nonidentity Element of a Group has Order 2, then it’s an Abelian Group Let $G$ be a group. Suppose that the order of nonidentity element of $G$ is $2$. Then show that $G$ is an abelian group.   Proof. Let $x$ and $y$ be elements of $G$. Then we have \[1=(xy)^2=(xy)(xy).\] Multiplying the equality by $yx$ from the right, we […]
  • A Recursive Relationship for a Power of a MatrixA Recursive Relationship for a Power of a Matrix Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation \[ A^{n} = b_n A + c_n I , \] where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]
  • Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian GroupTorsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian Group Let $A$ be an abelian group and let $T(A)$ denote the set of elements of $A$ that have finite order. (a) Prove that $T(A)$ is a subgroup of $A$. (The subgroup $T(A)$ is called the torsion subgroup of the abelian group $A$ and elements of $T(A)$ are called torsion […]

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