linear-algebra-eyecatch

• Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation $HE-EH=2E.$ Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]
• Sylow Subgroups of a Group of Order 33 is Normal Subgroups Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.   Hint. We use Sylow's theorem. Review the basic terminologies and Sylow's theorem. Recall that if there is only one $p$-Sylow subgroup $P$ of $G$ for a fixed prime $p$, then $P$ […]
• Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$ Let $G$ be a finite group of order $231=3\cdot 7 \cdot 11$. Prove that every Sylow $11$-subgroup of $G$ is contained in the center $Z(G)$. Hint. Prove that there is a unique Sylow $11$-subgroup of $G$, and consider the action of $G$ on the Sylow $11$-subgroup by […]
• An Example of Matrices $A$, $B$ such that $\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B)$ For an $m\times n$ matrix $A$, we denote by $\mathrm{rref}(A)$ the matrix in reduced row echelon form that is row equivalent to $A$. For example, consider the matrix $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &2 &2 \end{bmatrix}$ Then we have $A=\begin{bmatrix} 1 & 1 & 1 \\ […] • A Condition that a Commutator Group is a Normal Subgroup Let H be a normal subgroup of a group G. Then show that N:=[H, G] is a subgroup of H and N \triangleleft G. Here [H, G] is a subgroup of G generated by commutators [h,k]:=hkh^{-1}k^{-1}. In particular, the commutator subgroup [G, G] is a normal subgroup of […] • True or False Problems of Vector Spaces and Linear Transformations These are True or False problems. For each of the following statements, determine if it contains a wrong information or not. Let A be a 5\times 3 matrix. Then the range of A is a subspace in \R^3. The function f(x)=x^2+1 is not in the vector space C[-1,1] because […] • 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 […]