# Johns-Hopkins-University-exam-eye-catch

by Yu ·

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- 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, […]
- Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector. Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$. Solution. We will use the matrix components $A = (a_{i j})_{1 \leq i, j \leq n}$, $B = […]
- Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix \[A=\begin{bmatrix} 1 & 0 & 2 \\ 0 &3 &0 \\ 4 & 0 & 5 \end{bmatrix}.\] (a) Prove that the linear transformation […]
- Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible? A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix. Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of […]
- Normalize Lengths to Obtain an Orthonormal Basis Let \[ \mathbf{v}_{1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} ,\; \mathbf{v}_{2} = \begin{bmatrix} 1 \\ -1 \end{bmatrix} . \] Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$? If […]
- Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix. Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
- Find a Polynomial Satisfying the Given Conditions on Derivatives Find a cubic polynomial \[p(x)=a+bx+cx^2+dx^3\] such that $p(1)=1, p'(1)=5, p(-1)=3$, and $ p'(-1)=1$. Solution. By differentiating $p(x)$, we obtain \[p'(x)=b+2cx+3dx^2.\] Thus the given conditions are […]
- Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$. Let \[A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}.\] Find the eigenvalues and the eigenvectors of the matrix \[B=A^4-3A^3+3A^2-2A+8E.\] (Nagoya University Linear Algebra Exam Problem) Hint. Apply the Cayley-Hamilton theorem. That is if $p_A(t)$ is the […]