# field-theory-eyecatch

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- Is there an Odd Matrix Whose Square is $-I$? Let $n$ be an odd positive integer. Determine whether there exists an $n \times n$ real matrix $A$ such that \[A^2+I=O,\] where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix. If such a matrix $A$ exists, find an example. If not, prove that […]
- Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent. \begin{align*} S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \end{align*} where \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, […]
- Determine Whether Given Subsets in $\R^4$ are Subspaces or Not (a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying \[2x+4y+3z+7w+1=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a […]
- Determine Whether Trigonometry Functions $\sin^2(x), \cos^2(x), 1$ are Linearly Independent or Dependent Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$. Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent. (The Ohio State University, Linear Algebra Midterm Exam […]
- Show that Two Fields are Equal: $\Q(\sqrt{2}, \sqrt{3})= \Q(\sqrt{2}+\sqrt{3})$ Show that fields $\Q(\sqrt{2}+\sqrt{3})$ and $\Q(\sqrt{2}, \sqrt{3})$ are equal. Proof. It follows from $\sqrt{2}+\sqrt{3} \in \Q(\sqrt{2}, \sqrt{3})$ that we have $\Q(\sqrt{2}+\sqrt{3})\subset \Q(\sqrt{2}, \sqrt{3})$. To show the reverse inclusion, […]
- Every Finitely Generated Subgroup of Additive Group $\Q$ of Rational Numbers is Cyclic Let $\Q=(\Q, +)$ be the additive group of rational numbers. (a) Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic. (b) Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups. Proof. (a) Prove that every finitely generated […]
- Abelian Group and Direct Product of Its Subgroups Let $G$ be a finite abelian group of order $mn$, where $m$ and $n$ are relatively prime positive integers. Then show that there exists unique subgroups $G_1$ of order $m$ and $G_2$ of order $n$ such that $G\cong G_1 \times G_2$. Hint. Consider […]
- Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient. Let $W$ be the following subset of $P_3$. \[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\] Here $p'(x)$ is the first derivative of $p(x)$ and […]