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• Is there an Odd Matrix Whose Square is $-I$? Let $n$ be an odd positive integer. Determine whether there exists an $n \times n$ real matrix $A$ such that \[A^2+I=O,\] where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix. If such a matrix $A$ exists, find an example. If not, prove that […]
• Quiz 8. Determine Subsets are Subspaces: Functions Taking Integer Values / Set of Skew-Symmetric Matrices (a) Let $C[-1,1]$ be the vector space over $\R$ of all real-valued continuous functions defined on the interval $[-1, 1]$. Consider the subset $F$ of $C[-1, 1]$ defined by \[F=\{ f(x)\in C[-1, 1] \mid f(0) \text{ is an integer}\}.\] Prove or disprove that $F$ is a subspace of […]
• A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix Prove that the matrix \[A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}\] is diagonalizable. Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix. That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]
• Solve Linear Recurrence Relation Using Linear Algebra (Eigenvalues and Eigenvectors) Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation \[a_{k+2}-5a_{k+1}+3a_{k}=0\] for $k=1, 2, \dots$. Let $T$ be […]
• Find the Inverse Matrix of a Matrix With Fractions Find the inverse matrix of the matrix \[A=\begin{bmatrix} \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt] \frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt] -\frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{bmatrix}.\]   Hint. You may use the augmented matrix […]
• Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$ For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$. (a) $A=\begin{bmatrix} 4 & 2\\ 2& […]
• Differentiating Linear Transformation is Nilpotent Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less. Consider the differentiation linear transformation $T: P_n\to P_n$ defined by \[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\] (a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
• $\sqrt[m]{2}$ is an Irrational Number Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.   Hint. Use ring theory: Consider the polynomial $f(x)=x^m-2$. Apply Eisenstein's criterion, show that $f(x)$ is irreducible over $\Q$. Proof. Consider the monic polynomial […]