# Prime-Ideal

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• 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 […]
• Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup Let $G$ be a group and $H$ and $K$ be subgroups of $G$. For $h \in H$, and $k \in K$, we define the commutator $[h, k]:=hkh^{-1}k^{-1}$. Let $[H,K]$ be a subgroup of $G$ generated by all such commutators. Show that if $H$ and $K$ are normal subgroups of $G$, then the subgroup […]
• The Set $\{ a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$ is a Subspace in $C(\R)$ Let $C(\mathbb{R})$ be the vector space of real-valued functions on $\mathbb{R}$. Consider the set of functions $W = \{ f(x) = a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$. Prove that $W$ is a vector subspace of $C(\mathbb{R})$.   Proof. We verify […]
• Find a Value of a Linear Transformation From $\R^2$ to $\R^3$ Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are unit vectors of $\R^2$ and […]
• The Intersection of Two Subspaces is also a Subspace Let $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$. Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.   Definition (Intersection). Recall that the intersection $U\cap V$ is the set of elements that are both elements of $U$ […]
• Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities (a) Let $A=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 &1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}.$ Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue. (b) Let $A=\begin{bmatrix} 0 & 0 & 0 & 0 […] • True of False Problems on Determinants and Invertible Matrices Determine whether each of the following statements is True or False. (a) If A and B are n \times n matrices, and P is an invertible n \times n matrix such that A=PBP^{-1}, then \det(A)=\det(B). (b) If the characteristic polynomial of an n \times n matrix A […] • Diagonalize a 2 by 2 Matrix A and Calculate the Power A^{100} Let \[A=\begin{bmatrix} 1 & 2\\ 4& 3 \end{bmatrix}.$ (a) Find eigenvalues of the matrix $A$. (b) Find eigenvectors for each eigenvalue of $A$. (c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that […]