Are these vectors in the Nullspace of the Matrix?
Let $A=\begin{bmatrix}
1 & 0 & 3 & -2 \\
0 &3 & 1 & 1 \\
1 & 3 & 4 & -1
\end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$.
(a) $\begin{bmatrix}
-3 \\
0 \\
1 \\
0
\end{bmatrix}$
[…]

The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$
For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, […]

A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Definitions/Hint.
Recall the relevant definitions.
Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]

Every Group of Order 72 is Not a Simple Group
Prove that every finite group of order $72$ is not a simple group.
Definition.
A group $G$ is said to be simple if the only normal subgroups of $G$ are the trivial group $\{e\}$ or $G$ itself.
Hint.
Let $G$ be a group of order $72$.
Use the Sylow's theorem and determine […]

Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities
(a) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\]
Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.
(b) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 […]

The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$.
Let
\[R=\left\{\, \begin{bmatrix}
a & b\\
0& a
\end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\]
Then the usual matrix addition and multiplication make $R$ an ring.
Let
\[J=\left\{\, \begin{bmatrix}
0 & b\\
0& 0
\end{bmatrix} […]

Perturbation of a Singular Matrix is Nonsingular
Suppose that $A$ is an $n\times n$ singular matrix.
Prove that for sufficiently small $\epsilon>0$, the matrix $A-\epsilon I$ is nonsingular, where $I$ is the $n \times n$ identity matrix.
Hint.
Consider the characteristic polynomial $p(t)$ of the matrix $A$.
Note […]