# vector-space

• Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$. $V:=\left\{ \quad\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \in \R^4 \quad \middle| \quad x_1-x_2+x_3-x_4=0 \quad\right\}.$ Find a basis of the subspace $V$ […]
• Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$ Let $A=\begin{bmatrix} 1 & 3\\ 2& 4 \end{bmatrix}.$ Then (a) Find all matrices $B=\begin{bmatrix} x & y\\ z& w \end{bmatrix}$ such that $AB=BA$. (b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that $AB=BA \text{ and } […] • Beautiful Formulas for pi=3.14… The number \pi is defined a s the ratio of a circle's circumference C to its diameter d: \[\pi=\frac{C}{d}.$ $\pi$ in decimal starts with 3.14... and never end. I will show you several beautiful formulas for $\pi$.   Art Museum of formulas for $\pi$ […]
• Unit Vectors and Idempotent Matrices A square matrix $A$ is called idempotent if $A^2=A$. (a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$. Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$. Prove that $P$ is an idempotent matrix. (b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]
• Find 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 […]
• Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite Suppose $A$ is a positive definite symmetric $n\times n$ matrix. (a) Prove that $A$ is invertible. (b) Prove that $A^{-1}$ is symmetric. (c) Prove that $A^{-1}$ is positive-definite. (MIT, Linear Algebra Exam Problem)   Proof. (a) Prove that $A$ is […]
• Find an Orthonormal Basis of the Range of a Linear Transformation Let $T:\R^2 \to \R^3$ be a linear transformation given by $T\left(\, \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \,\right) = \begin{bmatrix} x_1-x_2 \\ x_2 \\ x_1+ x_2 \end{bmatrix}.$ Find an orthonormal basis of the range of $T$. (The Ohio […]
• Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure? If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module. Can this action be extended to make $M$ into a $\Q$-module?   Proof. In general, we cannot extend a $\Z$-module into a $\Q$-module. We give a counterexample. Let $M=\Zmod{2}$ be the order […]