# Three-pieces1

• Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix Suppose the following information is known about a $3\times 3$ matrix $A$. $A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=6\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 1 \\ -1 \\ 1 […] • 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 […]
• Short Exact Sequence and Finitely Generated Modules Let $R$ be a ring with $1$. Let $0\to M\xrightarrow{f} M' \xrightarrow{g} M^{\prime\prime} \to 0 \tag{*}$ be an exact sequence of left $R$-modules. Prove that if $M$ and $M^{\prime\prime}$ are finitely generated, then $M'$ is also finitely generated.   […]
• Order of Product of Two Elements in a Group Let $G$ be a group. Let $a$ and $b$ be elements of $G$. If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample.   Proof. We claim that it is not true. As a […]
• Common Eigenvector of Two Matrices and Determinant of Commutator Let $A$ and $B$ be $n\times n$ matrices. Suppose that these matrices have a common eigenvector $\mathbf{x}$. Show that $\det(AB-BA)=0$. Steps. Write down eigenequations of $A$ and $B$ with the eigenvector $\mathbf{x}$. Show that AB-BA is singular. A matrix is […]
• Group Homomorphism Sends the Inverse Element to the Inverse Element Let $G, G'$ be groups. Let $\phi:G\to G'$ be a group homomorphism. Then prove that for any element $g\in G$, we have $\phi(g^{-1})=\phi(g)^{-1}.$     Definition (Group homomorphism). A map $\phi:G\to G'$ is called a group homomorphism […]
• The Inverse Matrix is Unique Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.   Hint. That the inverse matrix of $A$ is unique means that there is only one inverse matrix of $A$. (That's why we say "the" inverse matrix of $A$ and denote it by […]