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• Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$ Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$. Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.   Solution. Let $\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}$. Then we want $\mathbf{v} […] • Condition that Two Matrices are Row Equivalent We say that two m\times n matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. Let A and I be 2\times 2 matrices defined as follows. \[A=\begin{bmatrix} 1 & b\\ c& d \end{bmatrix}, \qquad […] • Find All Matrices Satisfying a Given Relation Let a and b be two distinct positive real numbers. Define matrices \[A:=\begin{bmatrix} 0 & a\\ a & 0 \end{bmatrix}, \,\, B:=\begin{bmatrix} 0 & b\\ b& 0 \end{bmatrix}.$ Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a […]
• The Order of $ab$ and $ba$ in a Group are the Same Let $G$ be a finite group. Let $a, b$ be elements of $G$. Prove that the order of $ab$ is equal to the order of $ba$. (Of course do not assume that $G$ is an abelian group.)   Proof. Let $n$ and $m$ be the order of $ab$ and $ba$, respectively. That is, $(ab)^n=e, […] • If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup Let G be a finite group. Suppose that p is a prime number that divides the order of G. Let N be a normal subgroup of G and let P be a p-Sylow subgroup of G. Show that if P is normal in N, then P is a normal subgroup of G. Hint. It follows from […] • The Matrix Exponential of a Diagonal Matrix For a square matrix M, its matrix exponential is defined by \[e^M = \sum_{i=0}^\infty \frac{M^k}{k!}.$ Suppose that $M$ is a diagonal matrix $M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & […] • Determine the Values of a so that W_a is a Subspace For what real values of a is the set \[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}$ a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?   Solution. The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]
• Diagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix $A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}$ is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]