Purdue-Algebra-Exam-eye-catch

Purdue-Algebra-Exam-eye-catch

LoadingAdd to solve later

Sponsored Links

Purdue University Abstract Algebra Exam Problems and Solutions


LoadingAdd to solve later

Sponsored Links

More from my site

  • Matrix Operations with TransposeMatrix Operations with Transpose Calculate the following expressions, using the following matrices: \[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}\] (a) $A B^\trans + \mathbf{v} […]
  • Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$ Let $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.) For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of […]
  • Matrices Satisfying the Relation $HE-EH=2E$Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation \[HE-EH=2E.\] Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]
  • Find a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the RankFind a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the Rank For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix. (a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$. (b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$. (c) $C […]
  • Quiz: Linear Equations and Matrix EntreisQuiz: Linear Equations and Matrix Entreis Do the following quiz about Linear Equations Matrix entries.   There are two questions. After completing the quiz, click View questions to see the solutions.
  • Two Subspaces Intersecting Trivially, and the Direct Sum of Vector Spaces.Two Subspaces Intersecting Trivially, and the Direct Sum of Vector Spaces. Let $V$ and $W$ be subspaces of $\R^n$ such that $V \cap W =\{\mathbf{0}\}$ and $\dim(V)+\dim(W)=n$. (a) If $\mathbf{v}+\mathbf{w}=\mathbf{0}$, where $\mathbf{v}\in V$ and $\mathbf{w}\in W$, then show that $\mathbf{v}=\mathbf{0}$ and $\mathbf{w}=\mathbf{0}$. (b) If $B_1$ is a […]
  • Given the Characteristic Polynomial, Find the Rank of the MatrixGiven the Characteristic Polynomial, Find the Rank of the Matrix Let $A$ be a square matrix and its characteristic polynomial is given by \[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\] Find the rank of $A$. (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the degree of the characteristic polynomial […]
  • Possibilities For the Number of Solutions for a Linear SystemPossibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) \[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right. \] where $a,b,c, d$ […]

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.