Given a Spanning Set of the Null Space of a Matrix, Find the Rank
Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
\[\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}, \text{ and } \begin{bmatrix}
1 \\
-1 \\
0
[…]

Determine Whether Given Subsets in $\R^4$ are Subspaces or Not
(a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}$ satisfying
\[2x+4y+3z+7w+1=0.\]
Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a […]

Finite Order Matrix and its Trace
Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that
(a) $|\tr(A)|\leq n$.
(b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$.
(c) $\tr(A)=n$ if and only if $A=I_n$.
Proof.
(a) […]

Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure?
If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module.
Can this action be extended to make $M$ into a $\Q$-module?
Proof.
In general, we cannot extend a $\Z$-module into a $\Q$-module.
We give a counterexample. Let $M=\Zmod{2}$ be the order […]

Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)
(a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.
Find $\det(A)$.
(b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.
(c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of […]