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 […]
For What Values of $a$, Is the Matrix Nonsingular?
Determine the values of a real number $a$ such that the matrix
\[A=\begin{bmatrix}
3 & 0 & a \\
2 &3 &0 \\
0 & 18a & a+1
\end{bmatrix}\]
is nonsingular.
Solution.
We apply elementary row operations and obtain:
\begin{align*}
A=\begin{bmatrix}
3 & 0 & a […]
Polynomial $x^4-2x-1$ is Irreducible Over the Field of Rational Numbers $\Q$
Show that the polynomial
\[f(x)=x^4-2x-1\]
is irreducible over the field of rational numbers $\Q$.
Proof.
We use the fact that $f(x)$ is irreducible over $\Q$ if and only if $f(x+a)$ is irreducible for any $a\in \Q$.
We prove that the polynomial $f(x+1)$ is […]
Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup
Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$.
Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.
Proof.
To show that $f(N)$ is normal, we show that $gf(N)g^{-1}=f(N)$ for any $g \in […]
The Subspace of Linear Combinations whose Sums of Coefficients are zero
Let $V$ be a vector space over a scalar field $K$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset
\[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […]
A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
(a) Prove that for sufficiently small positive real $\epsilon$, the equation […]
Add to solve later
Matrix Representation, Rank, and Nullity of a Linear Transformation $T:\R^2\to \R^3$
Let $T:\R^2 \to \R^3$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
3 \\
2
\end{bmatrix} \,\right)
=\begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix} \text{ and }
T\left(\, \begin{bmatrix}
4\\
3
\end{bmatrix} […]
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 […]
For What Values of $a$, Is the Matrix Nonsingular?
Determine the values of a real number $a$ such that the matrix
\[A=\begin{bmatrix}
3 & 0 & a \\
2 &3 &0 \\
0 & 18a & a+1
\end{bmatrix}\]
is nonsingular.
Solution.
We apply elementary row operations and obtain:
\begin{align*}
A=\begin{bmatrix}
3 & 0 & a […]
The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$
Let $F$ be a field and let
\[H(F)=\left\{\, \begin{bmatrix}
1 & a & b \\
0 &1 &c \\
0 & 0 & 1
\end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\]
be the Heisenberg group over $F$.
(The group operation of the Heisenberg group is matrix […]
The Subspace of Linear Combinations whose Sums of Coefficients are zero
Let $V$ be a vector space over a scalar field $K$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset
\[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […]
A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
(a) Prove that for sufficiently small positive real $\epsilon$, the equation […]
