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Module Theory problems and solutions


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  • Eckmann–Hilton Argument: Group Operation is a Group HomomorphismEckmann–Hilton Argument: Group Operation is a Group Homomorphism Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying \[\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}\] for any $g\in G$. Let $\mu: G\times G \to G$ be a map defined […]
  • Compute $A^5\mathbf{u}$ Using Linear CombinationCompute $A^5\mathbf{u}$ Using Linear Combination Let \[A=\begin{bmatrix} -4 & -6 & -12 \\ -2 &-1 &-4 \\ 2 & 3 & 6 \end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix} 6 \\ 5 \\ -3 \end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}, \quad \text{ and } […]
  • 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 […]
  • Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable MatrixDetermine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix Let $A$ be an $n\times n$ matrix with the characteristic polynomial \[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable. (a) Find the size of the matrix $A$. (b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue […]
  • Injective Group Homomorphism that does not have Inverse HomomorphismInjective Group Homomorphism that does not have Inverse Homomorphism Let $A=B=\Z$ be the additive group of integers. Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$. (a) Prove that $\phi$ is a group homomorphism. (b) Prove that $\phi$ is injective. (c) Prove that there does not exist a group homomorphism $\psi:B […]
  • Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or LessFind a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (a) […]
  • Linear Algebra Midterm 1 at the Ohio State University (2/3)Linear Algebra Midterm 1 at the Ohio State University (2/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 2 and contains […]
  • Nilpotent Matrix and Eigenvalues of the MatrixNilpotent Matrix and Eigenvalues of the Matrix An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix. Prove the followings. (a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero. (b) The matrix $A$ is nilpotent if and only if […]

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