Find a Matrix that Maps Given Vectors to Given Vectors
Suppose that a real matrix $A$ maps each of the following vectors
\[\mathbf{x}_1=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}, \mathbf{x}_2=\begin{bmatrix}
0 \\
1 \\
1
\end{bmatrix}, \mathbf{x}_3=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix} \]
into the […]

Automorphism Group of $\Q(\sqrt[3]{2})$ Over $\Q$.
Determine the automorphism group of $\Q(\sqrt[3]{2})$ over $\Q$.
Proof.
Let $\sigma \in \Aut(\Q(\sqrt[3]{2}/\Q)$ be an automorphism of $\Q(\sqrt[3]{2})$ over $\Q$.
Then $\sigma$ is determined by the value $\sigma(\sqrt[3]{2})$ since any element $\alpha$ of $\Q(\sqrt[3]{2})$ […]

Every Group of Order 20449 is an Abelian Group
Prove that every group of order $20449$ is an abelian group.
Outline of the Proof
Note that $20449=11^2 \cdot 13^2$.
Let $G$ be a group of order $20449$.
We prove by Sylow's theorem that there are a unique Sylow $11$-subgroup and a unique Sylow $13$-subgroup of […]

If the Quotient Ring is a Field, then the Ideal is Maximal
Let $R$ be a ring with unit $1\neq 0$.
Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$.
(Do not assume that the ring $R$ is commutative.)
Proof.
Let $I$ be an ideal of $R$ such that
\[M \subset I \subset […]

Every Finite Group Having More than Two Elements Has a Nontrivial Automorphism
Prove that every finite group having more than two elements has a nontrivial automorphism.
(Michigan State University, Abstract Algebra Qualifying Exam)
Proof.
Let $G$ be a finite group and $|G|> 2$.
Case When $G$ is a Non-Abelian Group
Let us first […]

Determine Whether Each Set is a Basis for $\R^3$
Determine whether each of the following sets is a basis for $\R^3$.
(a) $S=\left\{\, \begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix}, \begin{bmatrix}
2 \\
1 \\
-1
\end{bmatrix}, \begin{bmatrix}
-2 \\
1 \\
4
\end{bmatrix} […]

Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix
Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
\[W=\{M\in V \mid AM=MA\}.\]
That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.
Determine which matrices are in the subspace $W$ […]

Basis of Span in Vector Space of Polynomials of Degree 2 or Less
Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients.
Let
\[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\]
be the set of four vectors in $P_2$.
Then find a basis of the subspace $\Span(S)$ among the vectors in $S$.
(Linear […]