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- Common Eigenvector of Two Matrices $A, B$ is Eigenvector of $A+B$ and $AB$.
Let $\lambda$ be an eigenvalue of $n\times n$ matrices $A$ and $B$ corresponding to the same eigenvector $\mathbf{x}$.
(a) Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$.
(b) Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to […]
- The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal
Let $G$ and $G'$ be groups and let $f:G \to G'$ be a group homomorphism.
If $H'$ is a normal subgroup of the group $G'$, then show that $H=f^{-1}(H')$ is a normal subgroup of the group $G$.
Proof.
We prove that $H$ is normal in $G$. (The fact that $H$ is a subgroup […]
- If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$
Let $N$ be a normal subgroup of a group $G$.
Suppose that $G/N$ is an infinite cyclic group.
Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.
Hint.
Use the fourth (or Lattice) isomorphism theorem.
Proof. […]
- Even Perfect Numbers and Mersenne Prime Numbers
Prove that if $2^n-1$ is a Mersenne prime number, then
\[N=2^{n-1}(2^n-1)\]
is a perfect number.
On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.
Definitions.
In this post, a […]
- Find a Condition that a Vector be a Linear Combination
Let
\[\mathbf{v}=\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}
2 \\
-1 \\
2
\end{bmatrix}.\]
Find the necessary and […]
- Probability of Getting Two Red Balls From the Chosen Box
There are two boxes containing red and blue balls. Let us call the boxes Box A and Box B. Each box contains the same number of red and blue balls. More specifically, Box A has 5 red balls and 5 blue balls. Box B has 20 red balls and 20 blue balls. You choose one box. Then draw two […]
- Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
(a) Let $A=\begin{bmatrix}
1 & 3 & 0 & 0 \\
1 &3 & 1 & 2 \\
1 & 3 & 1 & 2
\end{bmatrix}$.
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.
(b) Find the rank and nullity of the matrix $A$ in part (a).
Solution.
(a) […]
- Polynomial $x^4-2x-1$ is Irreducible Over the Field of Rational Numbers $\Q$
Show that the polynomial
\[f(x)=x^4-2x-1\]
is irreducible over the field of rational numbers $\Q$.
Proof.
We use the fact that $f(x)$ is irreducible over $\Q$ if and only if $f(x+a)$ is irreducible for any $a\in \Q$.
We prove that the polynomial $f(x+1)$ is […]