Group-Theory2

LoadingAdd to solve later

Group Theory Problems and Solutions


LoadingAdd to solve later

Sponsored Links

More from my site

  • In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable. Show that the matrix $A=\begin{bmatrix} 1 & \alpha\\ 0& 1 \end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$. Comment. Remark that if $A$ is a square […]
  • Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix?Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix? Let $A$ be an $n \times n$ matrix. Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.   Solution. The answer is true. Recall that the transpose of a matrix is the sum of its diagonal entries. Also, note that the […]
  • Questions About the Trace of a MatrixQuestions About the Trace of a Matrix Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix. (a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]
  • The Intersection of Two Subspaces is also a SubspaceThe Intersection of Two Subspaces is also a Subspace Let $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$. Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.   Definition (Intersection). Recall that the intersection $U\cap V$ is the set of elements that are both elements of $U$ […]
  • Rotation Matrix in Space and its Determinant and EigenvaluesRotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.\] (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
  • Basic Properties of Characteristic GroupsBasic Properties of Characteristic Groups Definition (automorphism). An isomorphism from a group $G$ to itself is called an automorphism of $G$. The set of all automorphism is denoted by $\Aut(G)$. Definition (characteristic subgroup). A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi […]
  • Solve a System of Linear Equations by Gauss-Jordan EliminationSolve a System of Linear Equations by Gauss-Jordan Elimination Solve the following system of linear equations using Gauss-Jordan elimination. \begin{align*} 6x+8y+6z+3w &=-3 \\ 6x-8y+6z-3w &=3\\ 8y \,\,\,\,\,\,\,\,\,\,\,- 6w &=6 \end{align*}   We use the following notation. Elementary row operations. The […]
  • The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$ For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.