# normal-subgroup

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• A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}$ is diagonalizable. Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix. That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]
• Find All the Square Roots of a Given 2 by 2 Matrix Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$. Find all the square roots of the matrix $A=\begin{bmatrix} 2 & 2\\ 2& 2 \end{bmatrix}.$   Proof. Diagonalize $A$. We first diagonalize the matrix […]
• Determine Null Spaces of Two Matrices Let $A=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ -1 & -3 & -4 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ 5 & 3 & 3 \end{bmatrix}.$ Determine the null spaces of matrices $A$ and $B$.   Proof. The null space of the […]
• Is the Derivative Linear Transformation Diagonalizable? Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by $T( ax^2 + bx + c ) = 2ax + b .$ Is $T$ diagonalizable? If so, find a diagonal matrix which […]
• Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known Let $T: \R^2 \to \R^2$ be a linear transformation. Let $\mathbf{u}=\begin{bmatrix} 1 \\ 2 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 3 \\ 5 \end{bmatrix}$ be 2-dimensional vectors. Suppose that \begin{align*} T(\mathbf{u})&=T\left( \begin{bmatrix} 1 \\ […]
• Galois Extension $\Q(\sqrt{2+\sqrt{2}})$ of Degree 4 with Cyclic Group Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group.   Proof. Put $\alpha=\sqrt{2+\sqrt{2}}$. Then we have $\alpha^2=2+\sqrt{2}$. Taking square of $\alpha^2-2=\sqrt{2}$, we obtain […]
• Find the Limit of a Matrix Let $A=\begin{bmatrix} \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\ \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\ \frac{3}{7} & \frac{3}{7} & \frac{1}{7} \end{bmatrix}$ be $3 \times 3$ matrix. Find $\lim_{n \to \infty} A^n.$ (Nagoya University Linear […]
• Every Finitely Generated Subgroup of Additive Group $\Q$ of Rational Numbers is Cyclic Let $\Q=(\Q, +)$ be the additive group of rational numbers. (a) Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic. (b) Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.   Proof. (a) Prove that every finitely generated […]