Simple Commutative Relation on Matrices
Let $A$ and $B$ are $n \times n$ matrices with real entries.
Assume that $A+B$ is invertible. Then show that
\[A(A+B)^{-1}B=B(A+B)^{-1}A.\]
(University of California, Berkeley Qualifying Exam)
Proof.
Let $P=A+B$. Then $B=P-A$.
Using these, we express the given […]

Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors
Let $V$ be a vector space and $B$ be a basis for $V$.
Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.
Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […]

Null Space, Nullity, Range, Rank of a Projection Linear Transformation
Let $\mathbf{u}=\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]
(a) […]

Group of Order $pq$ is Either Abelian or the Center is Trivial
Let $G$ be a group of order $|G|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers.
Then show that $G$ is either abelian group or the center $Z(G)=1$.
Hint.
Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […]

Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components
Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector.
Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$.
Solution.
We will use the matrix components $A = (a_{i j})_{1 \leq i, j \leq n}$, $B = […]

Does an Extra Vector Change the Span?
Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set
\[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\]
still a spanning set for […]

Unit Vectors and Idempotent Matrices
A square matrix $A$ is called idempotent if $A^2=A$.
(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.
Prove that $P$ is an idempotent matrix.
(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]