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Linear Algebra exam problems and solutions at University of California, Berkeley


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  • Basic Exercise Problems in Module TheoryBasic Exercise Problems in Module Theory Let $R$ be a ring with $1$ and $M$ be a left $R$-module. (a) Prove that $0_Rm=0_M$ for all $m \in M$. Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$. To simplify the […]
  • Diagonalize a 2 by 2 Symmetric MatrixDiagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   Solution. The characteristic polynomial $p(t)$ of the matrix $A$ […]
  • 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 […]
  • If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian?If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian? Let $R$ be a commutative ring with $1$. Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$. Is it true that $A$ is also a Noetherian ring?   Proof. The answer is no. We give a counterexample. Let […]
  • Every Group of Order 72 is Not a Simple GroupEvery Group of Order 72 is Not a Simple Group Prove that every finite group of order $72$ is not a simple group. Definition. A group $G$ is said to be simple if the only normal subgroups of $G$ are the trivial group $\{e\}$ or $G$ itself. Hint. Let $G$ be a group of order $72$. Use the Sylow's theorem and determine […]
  • Diagonalize the 3 by 3 Matrix if it is DiagonalizableDiagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}\] is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
  • If a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero MatrixIf a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero Matrix Let $A$ be an $n\times n$ singular matrix. Then prove that there exists a nonzero $n\times n$ matrix $B$ such that \[AB=O,\] where $O$ is the $n\times n$ zero matrix.   Definition. Recall that an $n \times n$ matrix $A$ is called singular if the […]
  • Primary Ideals, Prime Ideals, and Radical IdealsPrimary Ideals, Prime Ideals, and Radical Ideals Let $R$ be a commutative ring with unity. A proper ideal $I$ of $R$ is called primary if whenever $ab \in I$ for $a, b\in R$, then either $a\in I$ or $b^n\in I$ for some positive integer $n$. (a) Prove that a prime ideal $P$ of $R$ is primary. (b) If $P$ is a prime ideal and […]

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