Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself
Let $p$ be a prime number. Let
\[G=\{z\in \C \mid z^{p^n}=1\} \]
be the group of $p$-power roots of $1$ in $\C$.
Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.
Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ […]

Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$
Let
\[A=\begin{bmatrix}
1 & 2\\
4& 3
\end{bmatrix}.\]
(a) Find eigenvalues of the matrix $A$.
(b) Find eigenvectors for each eigenvalue of $A$.
(c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that […]

Basis with Respect to Which the Matrix for Linear Transformation is Diagonal
Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by
\[T(ax+b)=(3a+b)x+a+3,\]
for any $ax+b\in P_1$.
(a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation […]

The Coordinate Vector for a Polynomial with respect to the Given Basis
Let $\mathrm{P}_3$ denote the set of polynomials of degree $3$ or less with real coefficients. Consider the ordered basis
\[B = \left\{ 1+x , 1+x^2 , x - x^2 + 2x^3 , 1 - x - x^2 \right\}.\]
Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis […]

No Finite Abelian Group is Divisible
A nontrivial abelian group $A$ is called divisible if for each element $a\in A$ and each nonzero integer $k$, there is an element $x \in A$ such that $x^k=a$.
(Here the group operation of $A$ is written multiplicatively. In additive notation, the equation is written as $kx=a$.) That […]

Symmetric Matrix and Its Eigenvalues, Eigenspaces, and Eigenspaces
Let $A$ be a $4\times 4$ real symmetric matrix. Suppose that $\mathbf{v}_1=\begin{bmatrix}
-1 \\
2 \\
0 \\
-1
\end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $1$ of $A$.
Suppose that the eigenspace for the eigenvalue $2$ is $3$-dimensional.
(a) Find an […]

The Inverse Matrix of the Transpose is the Transpose of the Inverse Matrix
Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$.
Namely, show […]