Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector
Let
\[A=\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix}
1\\
0
\end{bmatrix}.\]
Compute $A^{2017}\mathbf{u}$.
(The Ohio State University, Linear Algebra Exam)
Solution.
We first compute $A\mathbf{u}$. We […]
Polynomial $x^p-x+a$ is Irreducible and Separable Over a Finite Field
Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements.
For any nonzero element $a\in \F_p$, prove that the polynomial
\[f(x)=x^p-x+a\]
is irreducible and separable over $F_p$.
(Dummit and Foote "Abstract Algebra" Section 13.5 Exercise #5 on […]
The Product of Two Nonsingular Matrices is Nonsingular
Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix.
(The Ohio State University, Linear Algebra Final Exam Problem)
Definition (Nonsingular Matrix)
An $n\times n$ matrix is called nonsingular if the […]
Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$
Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that
\[T(\mathbf{v}_1)=\begin{bmatrix}
1 \\
-2
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
3 \\
1 […]
Equivalent Conditions For a Prime Ideal in a Commutative Ring
Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent:
(a) The ideal $P$ is a prime ideal.
(b) For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.
Proof. […]
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 […]
Common Eigenvector of Two Matrices $A, B$ is Eigenvector of $A+B$ and $AB$.
Let $\lambda$ be an eigenvalue of $n\times n$ matrices $A$ and $B$ corresponding to the same eigenvector $\mathbf{x}$.
(a) Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$.
(b) Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to […]
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 $A=\begin{bmatrix}
a & b\\
c& d
\end{bmatrix}$ […]