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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 […]