# Harvard-University-exam-eye-catch

by Yu ·

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- If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial Let $x, y$ be generators of a group $G$ with relation \begin{align*} xy^2=y^3x,\tag{1}\\ yx^2=x^3y.\tag{2} \end{align*} Prove that $G$ is the trivial group. Proof. From the relation (1), we […]
- Simple Commutative Relation on Matrices Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that \[A(A+B)^{-1}B=B(A+B)^{-1}A.\] (University of California, Berkeley Qualifying Exam) Proof. Let $P=A+B$. Then $B=P-A$. Using these, we express the given […]
- If the Kernel of a Matrix $A$ is Trivial, then $A^T A$ is Invertible Let $A$ be an $m \times n$ real matrix. Then the kernel of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$. The kernel is also called the null space of $A$. Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is […]
- Find the Formula for the Power of a Matrix Let \[A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &0 &1 \\ 0 & 0 & 1 \end{bmatrix}\] be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$. Proof. We first compute several powers of $A$ and guess the general formula. We […]
- A Matrix Equation of a Symmetric Matrix and the Limit of its Solution Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$. (a) Prove that for sufficiently small positive real $\epsilon$, the equation […]
- Basis of Span in Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let \[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\] be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]
- Common Eigenvector of Two Matrices and Determinant of Commutator Let $A$ and $B$ be $n\times n$ matrices. Suppose that these matrices have a common eigenvector $\mathbf{x}$. Show that $\det(AB-BA)=0$. Steps. Write down eigenequations of $A$ and $B$ with the eigenvector $\mathbf{x}$. Show that AB-BA is singular. A matrix is […]
- Automorphism Group of $\Q(\sqrt[3]{2})$ Over $\Q$. Determine the automorphism group of $\Q(\sqrt[3]{2})$ over $\Q$. Proof. Let $\sigma \in \Aut(\Q(\sqrt[3]{2}/\Q)$ be an automorphism of $\Q(\sqrt[3]{2})$ over $\Q$. Then $\sigma$ is determined by the value $\sigma(\sqrt[3]{2})$ since any element $\alpha$ of $\Q(\sqrt[3]{2})$ […]