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Unique Factorization Domain Problems and Solutions


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  • Find the Nullspace and Range of the Linear Transformation $T(f)(x) = f(x)-f(0)$Find the Nullspace and Range of the Linear Transformation $T(f)(x) = f(x)-f(0)$ Let $C([-1, 1])$ denote the vector space of real-valued functions on the interval $[-1, 1]$. Define the vector subspace \[W = \{ f \in C([-1, 1]) \mid f(0) = 0 \}.\] Define the map $T : C([-1, 1]) \rightarrow W$ by $T(f)(x) = f(x) - f(0)$. Determine if $T$ is a linear map. If […]
  • Finite Group and a Unique Solution of an EquationFinite Group and a Unique Solution of an Equation Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that \[b^m=a.\]   We give two proofs. Proof 1. Since $m$ and $n$ are relatively prime […]
  • Matrix Representations for Linear Transformations of the Vector Space of PolynomialsMatrix Representations for Linear Transformations of the Vector Space of Polynomials Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less. Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$. For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation […]
  • Cosine and Sine Functions are Linearly IndependentCosine and Sine Functions are Linearly Independent Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$. Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.   Proof. Note that the zero vector in the vector space $C[-\pi, \pi]$ is […]
  • Determinant/Trace and Eigenvalues of a MatrixDeterminant/Trace and Eigenvalues of a Matrix Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues. Show that (1) $$\det(A)=\prod_{i=1}^n \lambda_i$$ (2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$ Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix […]
  • Determine a Condition on $a, b$ so that Vectors are Linearly DependentDetermine a Condition on $a, b$ so that Vectors are Linearly Dependent Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\ b \end{bmatrix}\] be vectors in $\R^3$. Determine a […]
  • If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal SubgroupIf Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup Let $H$ be a subgroup of a group $G$. Suppose that for each element $x\in G$, we have $x^2\in H$. Then prove that $H$ is a normal subgroup of $G$. (Purdue University, Abstract Algebra Qualifying Exam)   Proof. To show that $H$ is a normal subgroup of […]
  • Irreducible Polynomial $x^3+9x+6$ and Inverse Element in Field ExtensionIrreducible Polynomial $x^3+9x+6$ and Inverse Element in Field Extension Prove that the polynomial \[f(x)=x^3+9x+6\] is irreducible over the field of rational numbers $\Q$. Let $\theta$ be a root of $f(x)$. Then find the inverse of $1+\theta$ in the field $\Q(\theta)$.   Proof. Note that $f(x)$ is a monic polynomial and the prime […]

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