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MIT Linear Algebra Exam problems and solutions


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  • Are Groups of Order 100, 200 Simple?Are Groups of Order 100, 200 Simple? Determine whether a group $G$ of the following order is simple or not. (a) $|G|=100$. (b) $|G|=200$.   Hint. Use Sylow's theorem and determine the number of $5$-Sylow subgroup of the group $G$. Check out the post Sylow’s Theorem (summary) for a review of Sylow's […]
  • True of False Problems on Determinants and Invertible MatricesTrue of False Problems on Determinants and Invertible Matrices Determine whether each of the following statements is True or False. (a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$. (b) If the characteristic polynomial of an $n \times n$ matrix $A$ […]
  • If Two Ideals Are Comaximal in a Commutative Ring, then Their Powers Are Comaximal IdealsIf Two Ideals Are Comaximal in a Commutative Ring, then Their Powers Are Comaximal Ideals Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have \[I_1+I_2=R.\] Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.   > Proof. Since $I_1+I_2=R$, there exists $a \in I_1$ […]
  • Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$ Let $G$ be a group. Suppose that we have \[(ab)^3=a^3b^3\] for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$. Then prove that $G$ is an abelian group.   Proof. Let $a, b$ be arbitrary elements of the group $G$. We want […]
  • Linear Algebra Midterm 1 at the Ohio State University (3/3)Linear Algebra Midterm 1 at the Ohio State University (3/3) 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 3 and contains […]
  • Nilpotent Element a in a Ring and Unit Element $1-ab$Nilpotent Element a in a Ring and Unit Element $1-ab$ Let $R$ be a commutative ring with $1 \neq 0$. An element $a\in R$ is called nilpotent if $a^n=0$ for some positive integer $n$. Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$.   We give two proofs. Proof 1. Since $a$ […]
  • Compute Power of Matrix If Eigenvalues and Eigenvectors Are GivenCompute Power of Matrix If Eigenvalues and Eigenvectors Are Given Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where \[\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ […]
  • Find a Basis for the Range of a Linear Transformation of Vector Spaces of MatricesFind a Basis for the Range of a Linear Transformation of Vector Spaces of Matrices Let $V$ denote the vector space of $2 \times 2$ matrices, and $W$ the vector space of $3 \times 2$ matrices. Define the linear transformation $T : V \rightarrow W$ by \[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} a+b & 2d \\ 2b - d & -3c \\ 2b - c […]

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