Kyushu-University-Linear-Algebra

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Kyushu University Linear Algebra Exam Problems and Solutions


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  • Idempotent Matrix and its EigenvaluesIdempotent 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 NonsingularFind 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 $T(f)(x) = f(x)-f(0)$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 MatrixDeterminant 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 $AB$.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 $\R^2$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 IdealIn 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.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 […]

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