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- Every Prime Ideal in a PID is Maximal / A Quotient of a PID by a Prime Ideal is a PID (a) Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal. (b) Prove that a quotient ring of a PID by a prime ideal is a PID. Proof. (a) Prove that every PID is a maximal ideal. Let $R$ be a Principal Ideal Domain (PID) and let $P$ […]
- Subspaces of the Vector Space of All Real Valued Function on the Interval Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not. (a) $S=\{f(x) \in V \mid f(0)=f(1)\}$. (b) $T=\{f(x) \in V \mid […]
- Determine Whether Each Set is a Basis for $\R^3$ Determine whether each of the following sets is a basis for $\R^3$. (a) $S=\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} […]
- Torsion 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 […]
- Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known Let $T: \R^2 \to \R^2$ be a linear transformation. Let \[ \mathbf{u}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 3 \\ 5 \end{bmatrix}\] be 2-dimensional vectors. Suppose that \begin{align*} T(\mathbf{u})&=T\left( \begin{bmatrix} 1 \\ […]
- Powers of a Diagonal Matrix Let $A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$. Show that (1) $A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$ for any $n \in \N$. (2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix. Show that $B^n=S^{-1}A^n S$ for any $n \in […]
- Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent? A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. (a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent. (b) Let $P$ […]
- Does an Extra Vector Change the Span? Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set \[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\] still a spanning set for […]