# cody-slides

by Yu ·

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- If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial Let $x, y$ be generators of a group $G$ with relation \begin{align*} xy^2=y^3x,\tag{1}\\ yx^2=x^3y.\tag{2} \end{align*} Prove that $G$ is the trivial group. Proof. Let $e$ be the identity element of $G$. We […]
- A Matrix is Invertible If and Only If It is Nonsingular In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility. That is, we will prove that: A matrix $A$ is nonsingular if and only if $A$ is invertible. (a) Show that if $A$ is invertible, then $A$ is […]
- An Example of a Matrix that Cannot Be a Commutator Let $I$ be the $2\times 2$ identity matrix. Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$. Proof. Assume that $[A, B]=-I$. Then $ABA^{-1}B^{-1}=-I$ implies \[ABA^{-1}=-B. […]
- Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations Consider the matrix \[A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.\] (a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why. (b) Are the vectors \[ […]
- Is the Quotient Ring of an Integral Domain still an Integral Domain? Let $R$ be an integral domain and let $I$ be an ideal of $R$. Is the quotient ring $R/I$ an integral domain? Definition (Integral Domain). Let $R$ be a commutative ring. An element $a$ in $R$ is called a zero divisor if there exists $b\neq 0$ in $R$ such that […]
- Every Integral Domain Artinian Ring is a Field Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring. Prove that $R$ is a field. Definition (Artinian ring). A ring $R$ is called Artinian if it satisfies the defending chain condition on ideals. That is, whenever we have […]
- A Matrix Representation of a Linear Transformation and Related Subspaces Let $T:\R^4 \to \R^3$ be a linear transformation defined by \[ T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.\] (a) Find a matrix $A$ such that […]
- 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 […]