# ring-theory-eye-catch

by Yu · Published · Updated

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- Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix. Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
- Find All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular Find all the values of $x$ so that the following matrix $A$ is a singular matrix. \[A=\begin{bmatrix} x & x^2 & 1 \\ 2 &3 &1 \\ 0 & -1 & 1 \end{bmatrix}.\] Hint. Use the fact that a matrix is singular if and only if its determinant is […]
- Determine the Values of $a$ so that $W_a$ is a Subspace For what real values of $a$ is the set \[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\] a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions? Solution. The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]
- Possibilities of the Number of Solutions of a Homogeneous System of Linear Equations Here is a very short true or false problem. Select either True or False. Then click "Finish quiz" button. You will be able to see an explanation of the solution by clicking "View questions" button.
- If Two Ideals Are Comaximal in a Commutative Ring, then Their Powers Are Comaximal Ideals Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have \[I_1+I_2=R.\] Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal. > Proof. Since $I_1+I_2=R$, there exists $a \in I_1$ […]
- Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices Let $A, B, C$ be the following $3\times 3$ matrices. \[A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & […]
- Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017. (a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed. […]
- 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 […]