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• Is the Given Subset of The Ring of Integer Matrices an Ideal? Let $R$ be the ring of all $2\times 2$ matrices with integer coefficients: $R=\left\{\, \begin{bmatrix} a & b\\ c& d \end{bmatrix} \quad \middle| \quad a, b, c, d\in \Z \,\right\}.$ Let $S$ be the subset of $R$ given by $S=\left\{\, \begin{bmatrix} s & […] • 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 […] • Example of an Infinite Algebraic Extension Find an example of an infinite algebraic extension over the field of rational numbers \Q other than the algebraic closure \bar{\Q} of \Q in \C. Definition (Algebraic Element, Algebraic Extension). Let F be a field and let E be an extension of […] • Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}$ is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
• 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. […] • Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices Let A, B, C be the following 3\times 3 matrices. \[A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & […] • Union of Two Subgroups is Not a Group Let G be a group and let H_1, H_2 be subgroups of G such that H_1 \not \subset H_2 and H_2 \not \subset H_1. (a) Prove that the union H_1 \cup H_2 is never a subgroup in G. (b) Prove that a group cannot be written as the union of two proper […] • Exponential Functions are Linearly Independent Let c_1, c_2,\dots, c_n be mutually distinct real numbers. Show that exponential functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}$ are linearly independent over $\R$. Hint. Consider a linear combination $a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.$ […]