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• Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities (a) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 &1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}.\] Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue. (b) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 […]
• Example of Two Groups and a Subgroup of the Direct Product that is Not of the Form of Direct Product Give an example of two groups $G$ and $H$ and a subgroup $K$ of the direct product $G\times H$ such that $K$ cannot be written as $K=G_1\times H_1$, where $G_1$ and $H_1$ are subgroups of $G$ and $H$, respectively.   Solution. Let $G$ be any nontrivial group, and let […]
• Are Linear Transformations of Derivatives and Integrations Linearly Independent? Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation). Let $V$ be the vector space of all linear transformations from $W$ to $W$. The addition and the scalar multiplication of $V$ […]
• Probability Problems about Two Dice Two fair and distinguishable six-sided dice are rolled. (1) What is the probability that the sum of the upturned faces will equal $5$? (2) What is the probability that the outcome of the second die is strictly greater than the first die? Solution. The sample space $S$ is […]
• Quiz 13 (Part 2) Find Eigenvalues and Eigenvectors of a Special Matrix Find all eigenvalues of the matrix \[A=\begin{bmatrix} 0 & i & i & i \\ i &0 & i & i \\ i & i & 0 & i \\ i & i & i & 0 \end{bmatrix},\] where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric […]
• No/Infinitely Many Square Roots of 2 by 2 Matrices (a) Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ does not have a square root. Namely, show that there is no complex matrix $B$ such that $B^2=A$. (b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root […]
• Every Diagonalizable Nilpotent Matrix is the Zero Matrix Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.   Definition (Nilpotent Matrix) A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$. Proof. Main Part Since $A$ is […]
• Null Space, Nullity, Range, Rank of a Projection Linear Transformation Let $\mathbf{u}=\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation \[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\] (a) […]