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Problems and solutions of ring theory in abstract algebra


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  • A Condition that a Vector is a Linear Combination of Columns Vectors of a MatrixA Condition that a Vector is a Linear Combination of Columns Vectors of a Matrix Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$. Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the […]
  • Find the Formula for the Power of a MatrixFind the Formula for the Power of a Matrix Let \[A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &0 &1 \\ 0 & 0 & 1 \end{bmatrix}\] be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.   Proof. We first compute several powers of $A$ and guess the general formula. We […]
  • A Matrix Having One Positive Eigenvalue and One Negative EigenvalueA Matrix Having One Positive Eigenvalue and One Negative Eigenvalue Prove that the matrix \[A=\begin{bmatrix} 1 & 1.00001 & 1 \\ 1.00001 &1 &1.00001 \\ 1 & 1.00001 & 1 \end{bmatrix}\] has one positive eigenvalue and one negative eigenvalue. (University of California, Berkeley Qualifying Exam Problem)   Solution. Let us put […]
  • 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 $\pi$ […]
  • Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)Trace, Determinant, and Eigenvalue (Harvard University Exam Problem) (a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$. Find $\det(A)$. (b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$. (c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of […]
  • Every Ideal of the Direct Product of Rings is the Direct Product of IdealsEvery Ideal of the Direct Product of Rings is the Direct Product of Ideals Let $R$ and $S$ be rings with $1\neq 0$. Prove that every ideal of the direct product $R\times S$ is of the form $I\times J$, where $I$ is an ideal of $R$, and $J$ is an ideal of $S$.   Proof. Let $K$ be an ideal of the direct product $R\times […]
  • Annihilator of a Submodule is a 2-Sided Ideal of a RingAnnihilator of a Submodule is a 2-Sided Ideal of a Ring Let $R$ be a ring with $1$ and let $M$ be a left $R$-module. Let $S$ be a subset of $M$. The annihilator of $S$ in $R$ is the subset of the ring $R$ defined to be \[\Ann_R(S)=\{ r\in R\mid rx=0 \text{ for all } x\in S\}.\] (If $rx=0, r\in R, x\in S$, then we say $r$ annihilates […]
  • Determine Whether the Following Matrix Invertible. If So Find  Its Inverse Matrix.Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix. Let A be the matrix \[\begin{bmatrix} 1 & -1 & 0 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}.\] Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse. (The Ohio State University Linear Algebra […]

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