Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2
(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
(b) Find all such matrices with rank 2.
Solution.
(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
First we look at the rank 1 case. […]
Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent
Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent.
\begin{align*}
S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\},
\end{align*}
where
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
3 \\
1
\end{bmatrix}, […]
Sylow’s Theorem (Summary) In this post we review Sylow's theorem and as an example we solve the following problem.
Show that a group of order $200$ has a normal Sylow $5$-subgroup.
Review of Sylow's Theorem
One of the important theorems in group theory is Sylow's theorem.
Sylow's theorem is a […]
Group Homomorphism Sends the Inverse Element to the Inverse Element
Let $G, G'$ be groups. Let $\phi:G\to G'$ be a group homomorphism.
Then prove that for any element $g\in G$, we have
\[\phi(g^{-1})=\phi(g)^{-1}.\]
Definition (Group homomorphism).
A map $\phi:G\to G'$ is called a group homomorphism […]
The Transpose of a Nonsingular Matrix is Nonsingular
Let $A$ be an $n\times n$ nonsingular matrix.
Prove that the transpose matrix $A^{\trans}$ is also nonsingular.
Definition (Nonsingular Matrix).
By definition, $A^{\trans}$ is a nonsingular matrix if the only solution to
[…]
The Vector $S^{-1}\mathbf{v}$ is the Coordinate Vector of $\mathbf{v}$
Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$.
Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible.
Prove that for each vector $\mathbf{v} \in V$, the vector […]
The Ideal Generated by a Non-Unit Irreducible Element in a PID is Maximal
Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element.
Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal.
Proof.
Suppose that we have an ideal $I$ of $R$ such that
\[(a) \subset I \subset […]