linear-algebra-eyecatch

LoadingAdd to solve later

Sponsored Links

Linear Algebra Problems and Solutions


LoadingAdd to solve later

Sponsored Links

More from my site

  • Matrices Satisfying the Relation $HE-EH=2E$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 SubgroupsSylow 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 $Z(G)$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 $\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B)$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 SubgroupA 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 TransformationsTrue 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 SeriesInfinite 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 […]
  • Determine Linearly Independent or Linearly Dependent. Express as a Linear CombinationDetermine Linearly Independent or Linearly Dependent. Express as a Linear Combination Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others. \[\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 […]

Leave a Reply

Your email address will not be published. Required fields are marked *