# Purdue-univeristy-exam-eye-catch

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- How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}\] by finding a nonsingular […]
- Find a Basis for a Subspace of the Vector Space of $2\times 2$ Matrices Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where \begin{align*} A_1=\begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}, \quad A_2=\begin{bmatrix} 0 & -1 \\ 1 & 4 […]
- Equation $x_1^2+\cdots +x_k^2=-1$ Doesn’t Have a Solution in Number Field $\Q(\sqrt[3]{2}e^{2\pi i/3})$ Let $\alpha= \sqrt[3]{2}e^{2\pi i/3}$. Prove that $x_1^2+\cdots +x_k^2=-1$ has no solutions with all $x_i\in \Q(\alpha)$ and $k\geq 1$. Proof. Note that $\alpha= \sqrt[3]{2}e^{2\pi i/3}$ is a root of the polynomial $x^3-2$. The polynomial $x^3-2$ is […]
- Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$ Let $G$ be a group. Suppose that we have \[(ab)^3=a^3b^3\] for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$. Then prove that $G$ is an abelian group. Proof. Let $a, b$ be arbitrary elements of the group $G$. We want […]
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