# Three-pieces1

• Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism? Let $V$ denote the vector space of all real $2\times 2$ matrices. Suppose that the linear transformation from $V$ to $V$ is given as below. $T(A)=\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}A-A\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}.$ Prove or […]
• An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$. A linear transformation $T:\R^n \to \R^n$ is called orthogonal transformation if for all $\mathbf{x}, \mathbf{y}\in […] • Powers of a Diagonal Matrix Let$A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$. Show that (1)$A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$for any$n \in \N$. (2) Let$B=S^{-1}AS$, where$S$be an invertible$2 \times 2$matrix. Show that$B^n=S^{-1}A^n S$for any$n \in […]
• Any Vector is a Linear Combination of Basis Vectors Uniquely Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as $\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,$ where $c_1, c_2, c_3$ are […]
• 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 […]
• Infinite Cyclic Groups Do Not Have Composition Series Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.   Proof. Let $G=\langle a \rangle$ and suppose that $G$ has a composition series $G=G_0\rhd G_1 \rhd \cdots G_{m-1} \rhd G_m=\{e\},$ where $e$ is the identity element of […]
• Is the Set of Nilpotent Element an Ideal? Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$? If so, prove it. Otherwise give a counterexample.   Proof. We give a counterexample. Let $R$ be the noncommutative ring of $2\times 2$ matrices with real […]
• Example of an Element in the Product of Ideals that Cannot be Written as the Product of Two Elements Let $I=(x, 2)$ and $J=(x, 3)$ be ideal in the ring $\Z[x]$. (a) Prove that $IJ=(x, 6)$. (b) Prove that the element $x\in IJ$ cannot be written as $x=f(x)g(x)$, where $f(x)\in I$ and $g(x)\in J$.   Hint. If $I=(a_1,\dots, a_m)$ and $J=(b_1, \dots, b_n)$ are […]