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- Find All Symmetric Matrices satisfying the Equation Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$? Express your solution using free variable(s). Solution. Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ […]
- Basis and Dimension of the Subspace of All Polynomials of Degree 4 or Less Satisfying Some Conditions. Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients. Let $W$ be the subspace of $P_2$ by \[W=\{ p(x)\in P_4 \mid p(1)+p(-1)=0 \text{ and } p(2)+p(-2)=0 \}.\] Find a basis of the subspace $W$ and determine the dimension of […]
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- Is the Quotient Ring of an Integral Domain still an Integral Domain? Let $R$ be an integral domain and let $I$ be an ideal of $R$. Is the quotient ring $R/I$ an integral domain? Definition (Integral Domain). Let $R$ be a commutative ring. An element $a$ in $R$ is called a zero divisor if there exists $b\neq 0$ in $R$ such that […]
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- Prove a Group is Abelian if $(ab)^2=a^2b^2$ Let $G$ be a group. Suppose that \[(ab)^2=a^2b^2\] for any elements $a, b$ in $G$. Prove that $G$ is an abelian group. Proof. To prove that $G$ is an abelian group, we need \[ab=ba\] for any elements $a, b$ in $G$. By the given […]
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