Michigan-State-University-Abstract-Algebra-eye-catch

Michigan-State-University-Abstract-Algebra-eye-catch

LoadingAdd to solve later

Sponsored Links

Michigan State University Abstract Algebra Exam Problems and Solutions


LoadingAdd to solve later

Sponsored Links

More from my site

  • If a Power of a Matrix is the Identity, then the Matrix is DiagonalizableIf a Power of a Matrix is the Identity, then the Matrix is Diagonalizable Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix. Show that the matrix $A$ is diagonalizable. Hint. Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is […]
  • Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$ Let \[A=\begin{bmatrix} 1 & 3\\ 2& 4 \end{bmatrix}.\] Then (a) Find all matrices \[B=\begin{bmatrix} x & y\\ z& w \end{bmatrix}\] such that $AB=BA$. (b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that \[AB=BA \text{ and } […]
  • A Matrix Similar to a Diagonalizable Matrix is Also DiagonalizableA Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.   Definitions/Hint. Recall the relevant definitions. Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
  • A Group Homomorphism that Factors though Another GroupA Group Homomorphism that Factors though Another Group Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$. Define a map $\bar{f}:H\to K$ as follows. For each […]
  • Beautiful Formulas for pi=3.14…Beautiful Formulas for pi=3.14… The number $\pi$ is defined a s the ratio of a circle's circumference $C$ to its diameter $d$: \[\pi=\frac{C}{d}.\] $\pi$ in decimal starts with 3.14... and never end. I will show you several beautiful formulas for $\pi$.   Art Museum of formulas for $\pi$ […]
  • Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis. Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$. (a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$. (b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of […]
  • Union of Two Subgroups is Not a GroupUnion 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 […]
  • Determine the Values of $a$ so that $W_a$ is a SubspaceDetermine the Values of $a$ so that $W_a$ is a Subspace For what real values of $a$ is the set \[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\] a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?   Solution. The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ 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.