Prime-Ideal

• Invertible Matrix Satisfying a Quadratic Polynomial Let $A$ be an $n \times n$ matrix satisfying $A^2+c_1A+c_0I=O,$ where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix. Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular). How about the converse? […]
• Commutator Subgroup and Abelian Quotient Group Let $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$. Let $N$ be a subgroup of $G$. Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.   Definitions. Recall that for any $a, b \in G$, the […]
• Given the Characteristic Polynomial, Find the Rank of the Matrix Let $A$ be a square matrix and its characteristic polynomial is give by $p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of $A$. (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the degree of the characteristic polynomial […]
• If Every Vector is Eigenvector, then Matrix is a Multiple of Identity Matrix Let $A$ be an $n\times n$ matrix. Assume that every vector $\mathbf{x}$ in $\R^n$ is an eigenvector for some eigenvalue of $A$. Prove that there exists $\lambda\in \R$ such that $A=\lambda I$, where $I$ is the $n\times n$ identity matrix.   Proof. Let us write […]
• If the Images of Vectors are Linearly Independent, then They Are Linearly Independent Let $T: \R^n \to \R^m$ be a linear transformation. Suppose that $S=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a subset of $\R^n$ such that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$. Prove that the set $S$ […]
• True or False Problems of Vector Spaces and Linear Transformations These are True or False problems. For each of the following statements, determine if it contains a wrong information or not. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because […]
• The Quadratic Integer Ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD) Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD). Proof. Any element of the ring $\Z[\sqrt{-5}]$ is of the form $a+b\sqrt{-5}$ for some integers $a, b$. The associated (field) norm $N$ is given […]