Sequence Converges to the Largest Eigenvalue of a Matrix
Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.
Furthermore, suppose that
\[|\lambda_1| > |\lambda_2| \geq \cdots \geq […]

Example of an Infinite Algebraic Extension
Find an example of an infinite algebraic extension over the field of rational numbers $\Q$ other than the algebraic closure $\bar{\Q}$ of $\Q$ in $\C$.
Definition (Algebraic Element, Algebraic Extension).
Let $F$ be a field and let $E$ be an extension of […]

Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space
(a) Let $A=\begin{bmatrix}
1 & 2 & 1 \\
3 &6 &4
\end{bmatrix}$ and let
\[\mathbf{a}=\begin{bmatrix}
-3 \\
1 \\
1
\end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix}
-2 \\
1 \\
0
\end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix}
1 \\
1 […]

Boolean Rings Do Not Have Nonzero Nilpotent Elements
Let $R$ be a commutative ring with $1$ such that every element $x$ in $R$ is idempotent, that is, $x^2=x$. (Such a ring is called a Boolean ring.)
(a) Prove that $x^n=x$ for any positive integer $n$.
(b) Prove that $R$ does not have a nonzero nilpotent […]

A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$
Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$.
Consider a linear transformation $T:U\to V$.
Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).
Hints.
You may use the folowing facts.
A linear […]

Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$
Consider the complex matrix
\[A=\begin{bmatrix}
\sqrt{2}\cos x & i \sin x & 0 \\
i \sin x &0 &-i \sin x \\
0 & -i \sin x & -\sqrt{2} \cos x
\end{bmatrix},\]
where $x$ is a real number between $0$ and $2\pi$.
Determine for which values of $x$ the […]

Transpose of a Matrix and Eigenvalues and Related Questions
Let $A$ be an $n \times n$ real matrix. Prove the followings.
(a) The matrix $AA^{\trans}$ is a symmetric matrix.
(b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.
(c) The matrix $AA^{\trans}$ is non-negative definite.
(An $n\times n$ […]

$\sqrt[m]{2}$ is an Irrational Number
Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.
Hint.
Use ring theory:
Consider the polynomial $f(x)=x^m-2$.
Apply Eisenstein's criterion, show that $f(x)$ is irreducible over $\Q$.
Proof.
Consider the monic polynomial […]