# Purdue-Algebra-Exam-eye-catch

by Yu ·

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- The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive Let $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers. Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive? If so, prove it. Otherwise, give a counterexample. Solution. The […]
- Infinite Cyclic Groups Do Not Have Composition Series Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series. Proof. Let $G=\langle a \rangle$ and suppose that $G$ has a composition series \[G=G_0\rhd G_1 \rhd \cdots G_{m-1} \rhd G_m=\{e\},\] where $e$ is the identity element of […]
- A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors Let $A$ be a singular $n\times n$ matrix. Let \[\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \dots, […]
- Non-Prime Ideal of Continuous Functions Let $R$ be the ring of all continuous functions on the interval $[0,1]$. Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$. Show that the set $I$ is an ideal of $R$ but is not a prime ideal. Proof. We first show that $I$ is an ideal of […]
- Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation Let $V$ be the vector space of all $n\times n$ real matrices. Let us fix a matrix $A\in V$. Define a map $T: V\to V$ by \[ T(X)=AX-XA\] for each $X\in V$. (a) Prove that $T:V\to V$ is a linear transformation. (b) Let $B$ be a basis of $V$. Let $P$ be the matrix […]
- 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 […]
- Every Maximal Ideal of a Commutative Ring is a Prime Ideal Let $R$ be a commutative ring with unity. Then show that every maximal ideal of $R$ is a prime ideal. We give two proofs. Proof 1. The first proof uses the following facts. Fact 1. An ideal $I$ of $R$ is a prime ideal if and only if $R/I$ is an integral […]