# find-eigenvalues-eigenvectors

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- How to Find Eigenvalues of a Specific Matrix. Find all eigenvalues of the following $n \times n$ matrix. \[ A=\begin{bmatrix} 0 & 0 & \cdots & 0 &1 \\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 &0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & […]
- The Additive Group of Rational Numbers and The Multiplicative Group of Positive Rational Numbers are Not Isomorphic Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers. Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups. Proof. Suppose, towards a […]
- Prove a Given Subset is a Subspace and Find a Basis and Dimension Let \[A=\begin{bmatrix} 4 & 1\\ 3& 2 \end{bmatrix}\] and consider the following subset $V$ of the 2-dimensional vector space $\R^2$. \[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\] (a) Prove that the subset $V$ is a subspace of $\R^2$. (b) Find a basis for […]
- Each Element in a Finite Field is the Sum of Two Squares Let $F$ be a finite field. Prove that each element in the field $F$ is the sum of two squares in $F$. Proof. Let $x$ be an element in $F$. We want to show that there exists $a, b\in F$ such that \[x=a^2+b^2.\] Since $F$ is a finite field, the characteristic $p$ of the field […]
- If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal Let $A$ be an $n\times n$ real skew-symmetric matrix. (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular. (b) Prove that \[B=(I-A)(I+A)^{-1}\] is an orthogonal matrix. Proof. (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular. The […]
- Differentiating Linear Transformation is Nilpotent Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less. Consider the differentiation linear transformation $T: P_n\to P_n$ defined by \[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\] (a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
- Rings $2\Z$ and $3\Z$ are Not Isomorphic Prove that the rings $2\Z$ and $3\Z$ are not isomorphic. Definition of a ring homomorphism. Let $R$ and $S$ be rings. A homomorphism is a map $f:R\to S$ satisfying $f(a+b)=f(a)+f(b)$ for all $a, b \in R$, and $f(ab)=f(a)f(b)$ for all $a, b \in R$. A […]
- Torsion Submodule, Integral Domain, and Zero Divisors Let $R$ be a ring with $1$. An element of the $R$-module $M$ is called a torsion element if $rm=0$ for some nonzero element $r\in R$. The set of torsion elements is denoted \[\Tor(M)=\{m \in M \mid rm=0 \text{ for some nonzero} r\in R\}.\] (a) Prove that if $R$ is an […]