# Kyushu-University-Linear-Algebra

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• 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 […]
• Find Values of $a$ so that the Matrix is Nonsingular Let $A$ be the following $3 \times 3$ matrix. $A=\begin{bmatrix} 1 & 1 & -1 \\ 0 &1 &2 \\ 1 & 1 & a \end{bmatrix}.$ Determine the values of $a$ so that the matrix $A$ is nonsingular.   Solution. We use the fact that a matrix is nonsingular if and only if […]
• Find the Nullspace and Range of the Linear Transformation $T(f)(x) = f(x)-f(0)$ Let $C([-1, 1])$ denote the vector space of real-valued functions on the interval $[-1, 1]$. Define the vector subspace $W = \{ f \in C([-1, 1]) \mid f(0) = 0 \}.$ Define the map $T : C([-1, 1]) \rightarrow W$ by $T(f)(x) = f(x) - f(0)$. Determine if $T$ is a linear map. If […]
• Determinant of a General Circulant Matrix Let $A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & […] • Common Eigenvector of Two Matrices A, B is Eigenvector of A+B and AB. Let \lambda be an eigenvalue of n\times n matrices A and B corresponding to the same eigenvector \mathbf{x}. (a) Show that 2\lambda is an eigenvalue of A+B corresponding to \mathbf{x}. (b) Show that \lambda^2 is an eigenvalue of AB corresponding to […] • Find Matrix Representation of Linear Transformation From \R^2 to \R^2 Let T: \R^2 \to \R^2 be a linear transformation such that \[T\left(\, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 4 \\ 1 \end{bmatrix}, T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 3 \\ 2 […] • In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal Let R be a principal ideal domain (PID) and let P be a nonzero prime ideal in R. Show that P is a maximal ideal in R. Definition A commutative ring R is a principal ideal domain (PID) if R is a domain and any ideal I is generated by a single element […] • How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}$ by finding a nonsingular […]