# Three-pieces2

• The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements Let $G$ be an abelian group. Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively. Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$. Also determine whether the statement is true if $G$ is a […]
• Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like. Consider the matrix $A=\begin{bmatrix} 3/2 & 2\\ -1& -3/2 \end{bmatrix} \in M_{2\times 2}(\R).$ (a) Find the eigenvalues and corresponding eigenvectors of $A$. (b) Show that for $\mathbf{v}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}\in \R^2$, we can choose […]
• Find a Condition that a Vector be a Linear Combination Let $\mathbf{v}=\begin{bmatrix} a \\ b \\ c \end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}.$ Find the necessary and […]
• Find a Matrix so that a Given Subset is the Null Space of the Matrix, hence it’s a Subspace Let $W$ be the subset of $\R^3$ defined by $W=\left \{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 5x_1-2x_2+x_3=0 \right \}.$ Exhibit a $1\times 3$ matrix $A$ such that $W=\calN(A)$, the null space of $A$. […]
• Sherman-Woodbery Formula for the Inverse Matrix Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies $\mathbf{v}^{\trans}\mathbf{u}\neq -1.$ Define the matrix […]
• No/Infinitely Many Square Roots of 2 by 2 Matrices (a) Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ does not have a square root. Namely, show that there is no complex matrix $B$ such that $B^2=A$. (b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root […]
• A Relation of Nonzero Row Vectors and Column Vectors Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that $\mathbf{y}A=\mathbf{y}.$ (Here a row vector means a $1\times n$ matrix.) Prove that there is a nonzero column vector $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{x}.$ (Here a […]
• Orthonormal Basis of Null Space and Row Space Let $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}$. (a) Find an orthonormal basis of the null space of $A$. (b) Find the rank of $A$. (c) Find an orthonormal basis of the row space of $A$. (The Ohio State University, Linear Algebra Exam […]