pi2018

LoadingAdd to solve later


LoadingAdd to solve later

More from my site

  • 12 Examples of Subsets that Are Not Subspaces of Vector Spaces12 Examples of Subsets that Are Not Subspaces of Vector Spaces Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace. (1) \[S_1=\left \{\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\] in […]
  • Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic PolynomialsEigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials Let $A$ be a $2\times 2$ real symmetric matrix. Prove that all the eigenvalues of $A$ are real numbers by considering the characteristic polynomial of $A$.   Proof. Let $A=\begin{bmatrix} a& b \\ c& d \end{bmatrix}$. Then […]
  • Construction of a Symmetric Matrix whose Inverse Matrix is ItselfConstruction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by \[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where […]
  • A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular MatrixA Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix Prove that the matrix \[A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}\] is diagonalizable. Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix. That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]
  • A Group of Order $20$ is SolvableA Group of Order $20$ is Solvable Prove that a group of order $20$ is solvable.   Hint. Show that a group of order $20$ has a unique normal $5$-Sylow subgroup by Sylow's theorem. See the post summary of Sylow’s Theorem to review Sylow's theorem. Proof. Let $G$ be a group of order $20$. The […]
  • Quiz 13 (Part 1) Diagonalize a MatrixQuiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
  • Nontrivial Action of a Simple Group on a Finite SetNontrivial Action of a Simple Group on a Finite Set Let $G$ be a simple group and let $X$ be a finite set. Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$. Then show that $G$ is a finite group and the order of $G$ divides $|X|!$. Proof. Since $G$ acts on $X$, it […]
  • Determinant of a General Circulant MatrixDeterminant of a General Circulant Matrix Let \[A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & […]

Leave a Reply

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