# kyoto-univerisity-exam-eye-catch

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• Using Gram-Schmidt Orthogonalization, Find an Orthogonal Basis for the Span Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors $\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$ if $\mathbf{w}_{1} = \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} ,\quad \mathbf{w}_{2} = \begin{bmatrix} 2 \\ -1 \\ […] • Ring of Gaussian Integers and Determine its Unit Elements Denote by i the square root of -1. Let \[R=\Z[i]=\{a+ib \mid a, b \in \Z \}$ be the ring of Gaussian integers. We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to $N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.$ Here $\bar{\alpha}$ is the complex conjugate of […]
• Basic Exercise Problems in Module Theory Let $R$ be a ring with $1$ and $M$ be a left $R$-module. (a) Prove that $0_Rm=0_M$ for all $m \in M$. Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$. To simplify the […]
• Jewelry Company Quality Test Failure Probability A jewelry company requires for its products to pass three tests before they are sold at stores. For gold rings, 90 % passes the first test, 85 % passes the second test, and 80 % passes the third test. If a product fails any test, the product is thrown away and it will not take the […]
• Degree of an Irreducible Factor of a Composition of Polynomials Let $f(x)$ be an irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be any polynomial in $F[x]$. Show that the degree of each irreducible factor of the composite polynomial $f(g(x))$ is divisible by $n$.   Hint. Use the following fact. Let $h(x)$ is an […]
• A Positive Definite Matrix Has a Unique Positive Definite Square Root Prove that a positive definite matrix has a unique positive definite square root.   In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem. After the proof, several extra problems about square […]
• Idempotent Matrix and its Eigenvalues Let $A$ be an $n \times n$ matrix. We say that $A$ is idempotent if $A^2=A$. (a) Find a nonzero, nonidentity idempotent matrix. (b) Show that eigenvalues of an idempotent matrix $A$ is either $0$ or $1$. (The Ohio State University, Linear Algebra Final Exam […]
• Hyperplane in $n$-Dimensional Space Through Origin is a Subspace A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors $\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n$ satisfying the linear equation of the form $a_1x_1+a_2x_2+\cdots+a_nx_n=b,$ […]

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