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• A Recursive Relationship for a Power of a Matrix Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation \[ A^{n} = b_n A + c_n I , \] where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]
• Idempotent Linear Transformation and Direct Sum of Image and Kernel Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$. We assume that $A$ is idempotent, that is, $A^2=A$. Then prove that \[\R^n=\im(T) \oplus \ker(T).\]   Proof. To prove the equality $\R^n=\im(T) […]
• Probability that Three Pieces Form a Triangle We have a stick of a unit length. Two points on the stick will be selected randomly (uniformly along the length of the stick) and independently. Then we break the stick at these two points so that we get three pieces of the stick. What is the probability that these three pieces form a […]
• Inner Product, Norm, and Orthogonal Vectors Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ are vectors in $\R^n$. Suppose that vectors $\mathbf{u}_1$, $\mathbf{u}_2$ are orthogonal and the norm of $\mathbf{u}_2$ is $4$ and $\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number $a$ in […]
• A Group of Linear Functions Define the functions $f_{a,b}(x)=ax+b$, where $a, b \in \R$ and $a>0$. Show that $G:=\{ f_{a,b} \mid a, b \in \R, a>0\}$ is a group . The group operation is function composition. Steps. Check one by one the followings. The group operation on $G$ is […]
• If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal Let $A$ and $B$ be $n\times n$ matrices. Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$. Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly […]
• Is the Quotient Ring of an Integral Domain still an Integral Domain? Let $R$ be an integral domain and let $I$ be an ideal of $R$. Is the quotient ring $R/I$ an integral domain?   Definition (Integral Domain). Let $R$ be a commutative ring. An element $a$ in $R$ is called a zero divisor if there exists $b\neq 0$ in $R$ such that […]
• Write a Vector as a Linear Combination of Three Vectors Write the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors \[\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \, \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} , \, \begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}.\]   Solution. We want to find […]