Kyushu-University-Linear-Algebra

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Kyushu University Linear Algebra Exam Problems and Solutions


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  • Prove Vector Space Properties Using Vector Space AxiomsProve Vector Space Properties Using Vector Space Axioms Using the axiom of a vector space, prove the following properties. Let $V$ be a vector space over $\R$. Let $u, v, w\in V$. (a) If $u+v=u+w$, then $v=w$. (b) If $v+u=w+u$, then $v=w$. (c) The zero vector $\mathbf{0}$ is unique. (d) For each $v\in V$, the additive inverse […]
  • Linear Independent Vectors and the Vector Space Spanned By ThemLinear Independent Vectors and the Vector Space Spanned By Them Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$. Let […]
  • Find the Limit of a MatrixFind the Limit of a Matrix Let \[A=\begin{bmatrix} \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\ \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\ \frac{3}{7} & \frac{3}{7} & \frac{1}{7} \end{bmatrix}\] be $3 \times 3$ matrix. Find \[\lim_{n \to \infty} A^n.\] (Nagoya University Linear […]
  • Vector Space of Polynomials and Coordinate VectorsVector Space of Polynomials and Coordinate Vectors Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]
  • Normal Nilpotent Matrix is Zero MatrixNormal Nilpotent Matrix is Zero Matrix A complex square ($n\times n$) matrix $A$ is called normal if \[A^* A=A A^*,\] where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$. A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero […]
  • Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional SpaceQuiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by \[W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.\] (a) Which of the following vectors are in the subset […]
  • Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$Prove the Ring Isomorphism $R[x,y]/(x) \cong R[y]$ Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$. Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$. Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.   Proof. Define the map $\psi: R[x,y] \to […]
  • Coordinate Vectors and Dimension of Subspaces (Span)Coordinate Vectors and Dimension of Subspaces (Span) Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$. Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of […]

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