# idempotent-matrix

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• A 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 […]
• Differentiation is a Linear Transformation Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients. (a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by $T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)$ for any $f(x)\in […] • If the Kernel of a Matrix$A$is Trivial, then$A^T A$is Invertible Let$A$be an$m \times n$real matrix. Then the kernel of$A$is defined as$\ker(A)=\{ x\in \R^n \mid Ax=0 \}$. The kernel is also called the null space of$A$. Suppose that$A$is an$m \times n$real matrix such that$\ker(A)=0$. Prove that$A^{\trans}A$is […] • Subspaces of the Vector Space of All Real Valued Function on the Interval Let$V$be the vector space over$\R$of all real valued functions defined on the interval$[0,1]$. Determine whether the following subsets of$V$are subspaces or not. (a)$S=\{f(x) \in V \mid f(0)=f(1)\}$. (b)$T=\{f(x) \in V \mid […]
• Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix Consider the matrix $A=\begin{bmatrix} a & -b\\ b& a \end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$. (a) Find all eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenspace $E_{\lambda}$. (c) Diagonalize the matrix $A$ by finding a […]
• Find All Symmetric Matrices satisfying the Equation Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$? Express your solution using free variable(s).   Solution. Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ […]
• Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.   Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
• Every Maximal Ideal of a Commutative Ring is a Prime Ideal Let $R$ be a commutative ring with unity. Then show that every maximal ideal of $R$ is a prime ideal.   We give two proofs. Proof 1. The first proof uses the following facts. Fact 1. An ideal $I$ of $R$ is a prime ideal if and only if $R/I$ is an integral […]