# Nagoya-university-eye-catch

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- 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 […]
- Matrix Representations for Linear Transformations of the Vector Space of Polynomials Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less. Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$. For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation […]
- Determine a Value of Linear Transformation From $\R^3$ to $\R^2$ Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that \[ T\left(\, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\,\right) =\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{ and }T\left(\, \begin{bmatrix} 0 \\ 1 \\ 1 […]
- Idempotent (Projective) Matrices are Diagonalizable Let $A$ be an $n\times n$ idempotent complex matrix. Then prove that $A$ is diagonalizable. Definition. An $n\times n$ matrix $A$ is said to be idempotent if $A^2=A$. It is also called projective matrix. Proof. In general, an $n \times n$ matrix $B$ is […]
- Group of Order $pq$ is Either Abelian or the Center is Trivial Let $G$ be a group of order $|G|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers. Then show that $G$ is either abelian group or the center $Z(G)=1$. Hint. Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […]
- Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017. (a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed. […]
- Galois Group of the Polynomial $x^2-2$ Let $\Q$ be the field of rational numbers. (a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$? (b) Find the Galois group of $f(x)$ over $\Q$. Solution. (a) The polynomial $f(x)=x^2-2$ is separable over $\Q$ The roots of the polynomial $f(x)$ are $\pm […]
- Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$ Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that \[T(\mathbf{v}_1)=\begin{bmatrix} 1 \\ -2 \end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix} 3 \\ 1 […]