Find a Quadratic Function Satisfying Conditions on Derivatives
Find a quadratic function $f(x) = ax^2 + bx + c$ such that $f(1) = 3$, $f'(1) = 3$, and $f^{\prime\prime}(1) = 2$.
Here, $f'(x)$ and $f^{\prime\prime}(x)$ denote the first and second derivatives, respectively.
Solution.
Each condition required on $f$ can be turned […]

Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible.
Let
\[A=\begin{bmatrix}
1 & 3 & 3 \\
-3 &-5 &-3 \\
3 & 3 & 1
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 4 & 3 \\
-4 &-6 &-3 \\
3 & 3 & 1
\end{bmatrix}.\]
For this problem, you may use the fact that both matrices have the same characteristic […]

Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors
Find a basis for $\Span(S)$ where $S=
\left\{
\begin{bmatrix}
1 \\ 2 \\ 1
\end{bmatrix}
,
\begin{bmatrix}
-1 \\ -2 \\ -1
\end{bmatrix}
,
\begin{bmatrix}
2 \\ 6 \\ -2
\end{bmatrix}
,
\begin{bmatrix}
1 \\ 1 \\ 3
\end{bmatrix}
\right\}$.
Solution.
We […]

Complex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues
Let $A$ be an $n\times n$ real matrix.
Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.
We give two proofs.
Proof 1.
Let $\mathbf{x}$ be an eigenvector corresponding to the […]

Isomorphism Criterion of Semidirect Product of Groups
Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.
The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation
\[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\]
where $a_i […]

Ring Homomorphisms and Radical Ideals
Let $R$ and $R'$ be commutative rings and let $f:R\to R'$ be a ring homomorphism.
Let $I$ and $I'$ be ideals of $R$ and $R'$, respectively.
(a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$.
(b) Prove that $\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$
(c) Suppose that $f$ is […]

A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix
Prove that the matrix
\[A=\begin{bmatrix}
0 & 1\\
-1& 0
\end{bmatrix}\]
is diagonalizable.
Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix.
That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]

How to Find Eigenvalues of a Specific Matrix.
Find all eigenvalues of the following $n \times n$ matrix.
\[
A=\begin{bmatrix}
0 & 0 & \cdots & 0 &1 \\
1 & 0 & \cdots & 0 & 0\\
0 & 1 & \cdots & 0 &0\\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & […]