An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism
Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$.
A linear transformation $T:\R^n \to \R^n$ is called orthogonal transformation if for all $\mathbf{x}, \mathbf{y}\in […]

Powers of a Diagonal Matrix
Let $A=\begin{bmatrix}
a & 0\\
0& b
\end{bmatrix}$.
Show that
(1) $A^n=\begin{bmatrix}
a^n & 0\\
0& b^n
\end{bmatrix}$ for any $n \in \N$.
(2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix.
Show that $B^n=S^{-1}A^n S$ for any $n \in […]

Any Vector is a Linear Combination of Basis Vectors Uniquely
Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as
\[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\]
where $c_1, c_2, c_3$ are […]

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 […]

Infinite Cyclic Groups Do Not Have Composition Series
Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.
Proof.
Let $G=\langle a \rangle$ and suppose that $G$ has a composition series
\[G=G_0\rhd G_1 \rhd \cdots G_{m-1} \rhd G_m=\{e\},\]
where $e$ is the identity element of […]

Is the Set of Nilpotent Element an Ideal?
Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$?
If so, prove it. Otherwise give a counterexample.
Proof.
We give a counterexample.
Let $R$ be the noncommutative ring of $2\times 2$ matrices with real […]