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Johns Hopkins University math exam problems and solutions


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  • Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$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 EquivalentCondition 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 RelationFind 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 SameThe 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 SubgroupIf 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 MatrixThe 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 SubspaceDetermine 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 DiagonalizableDiagonalize 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 […]

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