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  • The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD)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. Every element of the ring $\Z[\sqrt{5}]$ can be written as $a+b\sqrt{5}$ for some integers $a, b$. The (field) norm $N$ of an element $a+b\sqrt{5}$ is […]
  • 10 True or False Problems about Basic Matrix Operations10 True or False Problems about Basic Matrix Operations Test your understanding of basic properties of matrix operations. There are 10 True or False Quiz Problems. These 10 problems are very common and essential. So make sure to understand these and don't lose a point if any of these is your exam problems. (These are actual exam […]
  • 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 […]
  • Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8 Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$.   Proof. Let $G$ be a group of order $24$. Note that $24=2^3\cdot 3$. Let $P$ be a Sylow $2$-subgroup of $G$. Then $|P|=8$. Consider the action of the group $G$ on […]
  • The Preimage of Prime ideals are Prime IdealsThe Preimage of Prime ideals are Prime Ideals Let $f: R\to R'$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R'$. Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.   Proof. The preimage of an ideal by a ring homomorphism is an ideal. (See the post "The inverse image of an ideal by […]
  • Properties of Nonsingular and Singular MatricesProperties of Nonsingular and Singular Matrices An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$. Otherwise $A$ is called singular. (a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is […]
  • Vector Space of 2 by 2 Traceless MatricesVector Space of 2 by 2 Traceless Matrices Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers. Let \[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix} a & b\\ c& -a \end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of […]
  • If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set?If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set? Suppose that $A$ is a real $n\times n$ matrix. (a) Is it true that $A$ must commute with its transpose? (b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set. Is it true that the rows of $A$ must also form an orthonormal set? (University of […]

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