# cody-slides

cody-slides

• Find a Quadratic Function Satisfying Conditions on Derivatives Find a quadratic function $f(x) = ax^2 + bx + c$ such that $f(1) = 3$, $f'(1) = 3$, and $f^{\prime\prime}(1) = 2$. Here, $f'(x)$ and $f^{\prime\prime}(x)$ denote the first and second derivatives, respectively.   Solution. Each condition required on $f$ can be turned […]
• Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible. Let $A=\begin{bmatrix} 1 & 3 & 3 \\ -3 &-5 &-3 \\ 3 & 3 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 4 & 3 \\ -4 &-6 &-3 \\ 3 & 3 & 1 \end{bmatrix}.$ For this problem, you may use the fact that both matrices have the same characteristic […]
• Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors Find a basis for $\Span(S)$ where $S= \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} , \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix} , \begin{bmatrix} 2 \\ 6 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} \right\}$.   Solution. We […]
• Complex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues Let $A$ be an $n\times n$ real matrix. Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.   We give two proofs. Proof 1. Let $\mathbf{x}$ be an eigenvector corresponding to the […]
• Isomorphism Criterion of Semidirect Product of Groups Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism. The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation $(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),$ where $a_i […] • Ring Homomorphisms and Radical Ideals Let$R$and$R'$be commutative rings and let$f:R\to R'$be a ring homomorphism. Let$I$and$I'$be ideals of$R$and$R'$, respectively. (a) Prove that$f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that$\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$(c) Suppose that$f$is […] • 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 […] • How to Find Eigenvalues of a Specific Matrix. Find all eigenvalues of the following$n \times n\$ matrix. \[ A=\begin{bmatrix} 0 & 0 & \cdots & 0 &1 \\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 &0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & […]